From 95f290eb5d3fc41282b3bc8487e8cf6186063bd5 Mon Sep 17 00:00:00 2001 From: GildedJson Date: Thu, 16 Dec 2021 14:42:27 -0800 Subject: [PATCH 01/59] Initial commit --- .../bayesian_networks_representation/index.md | 1 + .../metadata.yml | 31 +++++++++++++++++++ 2 files changed, 32 insertions(+) create mode 100644 notebooks/bayesian_networks_representation/index.md create mode 100644 notebooks/bayesian_networks_representation/metadata.yml diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md new file mode 100644 index 00000000..5ab2f8a4 --- /dev/null +++ b/notebooks/bayesian_networks_representation/index.md @@ -0,0 +1 @@ +Hello \ No newline at end of file diff --git a/notebooks/bayesian_networks_representation/metadata.yml b/notebooks/bayesian_networks_representation/metadata.yml new file mode 100644 index 00000000..cbb7f907 --- /dev/null +++ b/notebooks/bayesian_networks_representation/metadata.yml @@ -0,0 +1,31 @@ +title: LN | Bayesian Networks Representation + +header: + title: Bayesian Networks Representation + description: + +authors: + label: + position: top + content: + - name: Mohammad Hosein Gheysarieh + role: Author + contact: + - icon: fab fa-github + link: https://github.com/mhgheisarieh + + - name: Mohammad Cheraghi + role: Author + contact: + - icon: fab fa-github + link: https://github.com/mohammadch80 + + - name: Seyyed Mohammad Sadegh Tabaian + role: Author + contact: + - icon: fab fa-github + link: https://github.com/GildedJson + +comments: + label: false + kind: comments \ No newline at end of file From 938c453339aba15f28bd90e4938b5ab3ccd9d844 Mon Sep 17 00:00:00 2001 From: GildedJson <60608967+GildedJson@users.noreply.github.com> Date: Fri, 17 Dec 2021 11:42:26 -0800 Subject: [PATCH 02/59] Changed addressing mode --- .../bayesian_networks_representation/index.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 64fd57d3..efc6df37 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -46,15 +46,15 @@ We use probabilistic models to reason about unknown variables, given evidences. Two variables are *independent* if: -![formula1](/assets/T_Formula1.PNG) +![formula1](assets/T_Formula1.PNG) If two variables are independent, their joint distribution is the product of their distributions. This can also be shown in another way: -![formula2](/assets/T_Formula2.PNG) +![formula2](assets/T_Formula2.PNG) We show independency as: -![formula3](/assets/T_Formula3.PNG) +![formula3](assets/T_Formula3.PNG) Note that real life joint distributions are at best close to independent. Independence is a *modelling assumption*. We'll get back to assumptions later. @@ -64,15 +64,15 @@ Unconditional (absolute) independence is very rare to come by, because the varia X is conditionally independent of Y given Z -![formula4](/assets/T_Formula4.PNG) +![formula4](assets/T_Formula4.PNG) if and only if -![formula5](/assets/T_Formula5.PNG) +![formula5](assets/T_Formula5.PNG) or equivalently, if and only if -![formula6](/assets/T_Formula6.PNG) +![formula6](assets/T_Formula6.PNG) ### Chain Rule @@ -110,4 +110,4 @@ or equivalently, if and only if ## Conclusion -## References \ No newline at end of file +## References From aae7f46c7968cd2fd9f31dd1c91c0ce3c46360d9 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 00:24:35 +0330 Subject: [PATCH 03/59] Update index.md check grammar --- .../bayesian_networks_representation/index.md | 42 +++++++++---------- 1 file changed, 21 insertions(+), 21 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 5d49375d..f1626cd3 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -184,9 +184,9 @@ is X is guaranteed to be independent of Z given Y? #### V-Structure -last configuration is "common effect" and sometimes called V-Structure. +the last configuration is "common effect" and is sometimes called V-Structure. -- in this configuration one variable is affected by another two variables. +- in this configuration, one variable is affected by another two variables. - P(x, y, z) = P(x) P(y) P(z|x, y) @@ -198,7 +198,7 @@ in following BN, both ballgame and raining cause traffic. is X is guaranteed to be independent of Y? -- the answer, unlike previous versions, is Yes. ballgame and raining do not get effect from each other or a common cause. to prove this claim, use algebraic operation again: +- the answer, unlike previous versions, is Yes. ballgame and rain do not get effect from each other or a common cause. to prove this claim, use algebraic operation again: ![ch_formula1](assets/ch12.PNG) @@ -206,19 +206,19 @@ is X is guaranteed to be independent of Y? is X is guaranteed to be independent of Y given z? -- the answer is NO. with an example we describe the situation. consider X and Y have random value and probability of occurrence each one is 0.5. (P(+x)=P(+y)=0.5) and traffic will occur when ballgame or raining occurs. +- the answer is NO. with an example we describe the situation. consider X and Y have random values and probability of occurrence each one is 0.5. (P(+x)=P(+y)=0.5) and traffic will occur when ballgame or raining occurs. - P(+z|+x,+y)=P(+z|+x,-y)=P(+z|-x,+y)=1 P(+z|-x,-y)=0 -- From problem description, We can conclude that P(+x|+z,-y)=1 and P(+x|-y)=0.5. So, P(X|Z,Y) is not equal to P(X|Y). +- From the problem description, We can conclude that P(+x|+z,-y)=1 and P(+x|-y)=0.5. So, P(X|Z,Y) is not equal to P(X|Y). -- So, with observation of Z, X and Y are not independent. this means Observing an effect, activates influence between possible causes. +- So, with the observation of Z, X and Y are not independent. this means Observing an effect, activating influence between possible causes. ### Reachability -Now, we want to use these three cases to check conditional dependency in any arbitrary graph. generally, the problem is checking dependency between two random variables in a BN in which some (or any) variables are observed. for this goal we break the graph into triples which we learned above and do some checking on them. In the following these steps are described. +Now, we want to use these three cases to check conditional dependency in any arbitrary graph. generally, the problem is checking dependency between two random variables in a BN in which some (or any) variables are observed. for this goal, we break the graph into triples which we learned above, and do some checking on them. In the following, these steps are described. -in the first step, we should shade evidence nodes, nodes that are observed in the problem. then looking for undirected path between determined random variables. the first idea is that if two nodes are connected by a path that is blocked by a shaded node, two random variables are independent. But there are drawbacks to this rule. when we have several paths between two RVs or there is V-structure triple that its bottom node is shaded, our method is wrong. So, we describe next step to correct these problems. +in the first step, we should shade evidence nodes, nodes that are observed in the problem. then looking for undirected paths between determined random variables. the first idea is that if two nodes are connected by a path that is blocked by a shaded node, two random variables are independent. But there are drawbacks to this rule. when we have several paths between two RVs or there is a V-structure triple that its bottom node is shaded, our method is wrong. So, we describe the next step to correct these problems. ### Active/Inactive Paths @@ -226,7 +226,7 @@ Question: are X and Y conditionally independent given evidence variables {Z}? - Yes, if x and y “d-separated” by z. -to describe "d-separated" concept, we first need to understand the concept of active/inactive path. +to describe the "d-separated" concept, we first need to understand the concept of active/inactive path. a path is active if each triple of it is active. a triple is active if: @@ -240,37 +240,37 @@ in the figure below, examples of active and inactive triples are shown. ![dsepr](assets/ch4.jpg) -note that a single inactive triple makes a path inactive. but two RVs are independent if all paths between related nodes are inactive. So, if some paths were active, independency is not guaranteed. +note that a single inactive triple makes a path inactive. but two RVs are independent if all paths between related nodes are inactive. So, if some paths were active, independence is not guaranteed. Example: -- a. we want to check conditionally independency between R and B variables with different evidence. +- a. we want to check conditionally independence between R and B variables with different evidence. ![exp1](assets/ch5.jpg) - **without evidence:** there is only one path between R and B that is a “common effect” triple. it is inactive so R and B are independent. -- **evidence=T:** there is a “common effect” triple that is active. So, independency of R and B aren’t guaranteed. +- **evidence=T:** there is a “common effect” triple that is active. So, the independence of R and B isn’t guaranteed. - **evidence=T’:** this situation is similar to previous evidence. -- b. we want to check conditionally independency between L and B variables with different evidence. +- b. we want to check conditionally independence between L and B variables with different evidence. ![exp2](assets/ch6.jpg) - **without evidence:** there is only one path between L and B that has two triples. L->R->T is active but R->T<-B is inactive. So, this pass is inactive. So, L and B are independent. -- **evidence = T:** similar to previous segment there is only one path that has two triples. both are active. So, L and B aren't guaranteed to be independent given T. -- **evidence = T':** it is similar to similar to segment. -- **evidence = T, R:** similar to 2’nd segment the R->T<-B triple is active but L->R->T is inactive. So, L and B are independent given T and R. +- **evidence = T:** similar to the previous segment there is only one path that has two triples. both are active. So, L and B aren't guaranteed to be independent given T. +- **evidence = T':** it is similar to segment. +- **evidence = T, R:** similar to the 2’nd segment the R->T<-B triple is active but L->R->T is inactive. So, L and B are independent given T and R. -- c. we want to check conditionally independency between T and D variables with different evidence. +- c. we want to check conditionally independence between T and D variables with different evidence. ![exp3](assets/ch7.jpg) -- **without evidence:** There are two paths between T and D. upper path is an active “common cause” and lower path is an inactive “common effect”. So, L and T aren't guaranteed to be independent. -- **evidence = R:** There are two paths between T and D. upper path is an inactive “common cause” and lower path is an inactive “common effect”. So, L and T are independent given R. -- **evidence = R, S:** There are two paths between T and D. upper path is an inactive “common cause” and lower path is an active “common effect”. So, L and T aren't guaranteed to be independent. +- **without evidence:** There are two paths between T and D. upper path is an active “common cause” and the lower path is an inactive “common effect”. So, L and T aren't guaranteed to be independent. +- **evidence = R:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an inactive “common effect”. So, L and T are independent given R. +- **evidence = R, S:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an active “common effect”. So, L and T aren't guaranteed to be independent. ### Structure Implications ## Conclusion -## References \ No newline at end of file +## References From 5998775ded920615906a8c9176d423bdb23ecb36 Mon Sep 17 00:00:00 2001 From: GildedJson Date: Fri, 17 Dec 2021 17:56:05 -0800 Subject: [PATCH 04/59] Table of contents --- .../bayesian_networks_representation/index.md | 73 ++++++++++++++++++- 1 file changed, 72 insertions(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 5ab2f8a4..ad35b925 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -1 +1,72 @@ -Hello \ No newline at end of file +# **Bayesian Networks Representation** + +## **Table of contents:** +- [Intoduction](#Intoduction) + - [Probabilistic Models](#Probabilistic-Models) +- [Independence](#Independence) + - [Conditional Independence](#Conditional-Independence) + - [Chain Rule](#Chain-Rule) +- [Bayes' Nets](Bayes'-Nets) + - [Problems with joint distribution tables](#Problems-with-joint-distribution-tables) + - [Graphical Notation](#Graphical-Notation) + - [Semantics](#Semantics) + - [Probabilities in Bayes' Nets](#Probabilities-in-Bayes'-Nets) + - [Causality in Bayes' Nets](#Causality-in-Bayes'-Nets) + - [Space Efficiency](#Space-Efficiency) +- [Independence in Bayes' Nets](#Independence-in-Bayes'-Nets) + - [Independency Assumptions](#Independency-Assumptions) + - [D-separation](#D-separation) + - [Causal Chain](#Causal-Chain) + - [Common Cause](#Common-Cause) + - [V-Structure](#V-Structure) + - [Reachability](#Reachability) + - [Active/Inactive Paths](#Active/Inactive-Paths) + - [Structure Implications](#Structure-Implications) +- [Conclusion](#Conclusion) +- [References](#References) + +## Intoduction + +### Probabilistic Models + +## Independence + +### Conditional Independence + +### Chain Rule + +## Bayes' Nets + +### Problems with joint distribution tables + +### Graphical Notation + +### Semantics + +### Probabilities in Bayes' Nets + +### Causality in Bayes' Nets + +### Space Efficiency + +## Independence in Bayes' Nets + +### Independency Assumptions + +### D-separation + +#### Causal Chain + +#### Common Cause + +#### V-Structure + +### Reachability + +### Active/Inactive Paths + +### Structure Implications + +## Conclusion + +## References \ No newline at end of file From 8802c4f6a1c3b13696467eea001aa9e903495425 Mon Sep 17 00:00:00 2001 From: GildedJson Date: Fri, 17 Dec 2021 21:04:23 -0800 Subject: [PATCH 05/59] Intro --- .../assets/Fridge_net.PNG | Bin 0 -> 16138 bytes .../bayesian_networks_representation/index.md | 13 +++++++++++++ 2 files changed, 13 insertions(+) create mode 100644 notebooks/bayesian_networks_representation/assets/Fridge_net.PNG diff --git a/notebooks/bayesian_networks_representation/assets/Fridge_net.PNG b/notebooks/bayesian_networks_representation/assets/Fridge_net.PNG new file mode 100644 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b/notebooks/bayesian_networks_representation/index.md index ad35b925..3c6da4d0 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -27,8 +27,21 @@ ## Intoduction +Imagine this scenario. You want to know if the food in your fridge is going to spoil. You can hear your fridge humming. Does that change the probability of the food spoiling? What if you see that the fridge door is open? Are these evidences even related to the food spoiling? In this lecture note, we want to introduce a probabilistic model to study this kind of problems. + ### Probabilistic Models +So, what is a probabilistic model? A model is a simplification of the real world; it describes how portions of the world work. Since models are simplifications, they do not contain all the details and may not account for every variable or interaction between variables. As George E. P. Box, a British statistician one said: "All models are wrong; but some are useful." + +We use probabilistic models to reason about unknown variables, given evidences. This can be done in three ways: + +- Diagnostic inference + Means going from effects to causes, or in other words **explaining** the effects. For instance, given that the food spoiled, infer the probability of the fridge being broken. +- Causal inference + Means going from causes to effects, or in other words **predicting** the effects. For example, given that the power is cut, find the probability of the food spoiling. +- Intercausal inference + Means inferring probabilities between causes of a common effect, which is often called **explaining away**. For example, if we know that the food spoiled and the fridge door was open, the probability of the fridge being broken gets very low, although the door being open is independent of the fridge being broken. + ## Independence ### Conditional Independence From 2f392cee8807a6f39e35193447b5436d317bcf20 Mon Sep 17 00:00:00 2001 From: GildedJson Date: Fri, 17 Dec 2021 23:07:34 -0800 Subject: [PATCH 06/59] Until conditional independence --- .../assets/T_Formula1.PNG | Bin 0 -> 6808 bytes .../assets/T_Formula2.PNG | Bin 0 -> 5569 bytes .../assets/T_Formula3.PNG | Bin 0 -> 2000 bytes .../assets/T_Formula4.PNG | Bin 0 -> 2144 bytes .../assets/T_Formula5.PNG | Bin 0 -> 5731 bytes .../assets/T_Formula6.PNG | Bin 0 -> 4746 bytes .../bayesian_networks_representation/index.md | 28 ++++++++++++++++++ 7 files changed, 28 insertions(+) create mode 100644 notebooks/bayesian_networks_representation/assets/T_Formula1.PNG create mode 100644 notebooks/bayesian_networks_representation/assets/T_Formula2.PNG 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This can also be shown in another way: + +![formula2](/assets/T_Formula2.PNG) + +We show independency as: + +![formula3](/assets/T_Formula3.PNG) + +Note that real life joint distributions are at best close to independent. Independence is a *modelling assumption*. We'll get back to assumptions later. + ### Conditional Independence +Unconditional (absolute) independence is very rare to come by, because the variables in a problem are usually all correlated. Conditional independence is our most basic and robust form of knowledge about uncertain environments. + +X is conditionally independent of Y given Z + +![formula4](/assets/T_Formula4.PNG) + +if and only if + +![formula5](/assets/T_Formula5.PNG) + +or equivalently, if and only if + +![formula6](/assets/T_Formula6.PNG) + ### Chain Rule ## Bayes' Nets From bdff412055069d17cdfbeee24a35368c6a233a9b Mon Sep 17 00:00:00 2001 From: GildedJson Date: Sat, 18 Dec 2021 00:00:22 -0800 Subject: [PATCH 07/59] Added Cheraghi's parts --- .../assets/ch1.png | Bin 0 -> 149295 bytes .../assets/ch2.jpg | Bin 0 -> 42060 bytes .../assets/ch3.jpg | Bin 0 -> 38746 bytes .../assets/ch4.jpg | Bin 0 -> 40006 bytes .../assets/ch5.jpg | Bin 0 -> 11610 bytes .../assets/ch6.jpg | Bin 0 -> 21109 bytes .../assets/ch7.jpg | Bin 0 -> 12687 bytes .../bayesian_networks_representation/index.md | 175 +++++++++++++++++- 8 files changed, 169 insertions(+), 6 deletions(-) create mode 100644 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X to Y after shading evidence variables, if none of these paths aren’t active paths, we can say X and Y are d-separated by Z. + +in the figure below, examples of active and inactive triples are shown. + +![dsepr](assets/ch4.jpg) + +note that a single inactive triple makes a path inactive. but two RVs are independent if all paths between related nodes are inactive. So, if some paths were active, independency is not guaranteed. + +Example: + +- a. we want to check conditionally independency between R and B variables with different evidence. + +![exp1](assets/ch5.jpg) + + - **without evidence:** there is only one path between R and B that is a “common effect” triple. it is inactive so R and B are independent. + - **evidence=T:** there is a “common effect” triple that is active. So, independency of R and B aren’t guaranteed. + - **evidence=T’:** this situation is similar to previous evidence. + +- b. we want to check conditionally independency between L and B variables with different evidence. + +![exp2](assets/ch6.jpg) + + - **without evidence:** there is only one path between L and B that has two triples. L->R->T is active but R->T<-B is inactive. So, this pass is inactive. So, L and B are independent. + - **evidence = T:** similar to previous segment there is only one path that has two triples. both are active. So, L and B aren't guaranteed to be independent given T. + - **evidence = T':** it is similar to similar to segment. + - **evidence = T, R:** similar to 2’nd segment the R->T<-B triple is active but L->R->T is inactive. So, L and B are independent given T and R. + +- c. we want to check conditionally independency between T and D variables with different evidence. + +![exp3](assets/ch7.jpg) + + - **without evidence:** There are two paths between T and D. upper path is an active “common cause” and lower path is an inactive “common effect”. So, L and T aren't guaranteed to be independent. + - **evidence = R:** There are two paths between T and D. upper path is an inactive “common cause” and lower path is an inactive “common effect”. So, L and T are independent given R. + - **evidence = R, S:** There are two paths between T and D. upper path is an inactive “common cause” and lower path is an active “common effect”. 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zD6Scn`26gvzKw;%W9V$8jba`zW9e2S=U#?-#@IDFAPN zJ|JthL4GSzq6rn?1zo=gK$_%?j3eH;cAgIB0nP7%V0&=~2~I4g?E(D>fZ9(&D%OQW zGXq%E5qLI&qqydSXEodXPZDh9*x`cjwgHL?s67mDP~Jfj-@!sgCf+XZ0nP0Ksw$gF znv*)@7X;Ys@~0=VEChkmI{g!4{4aaDw}zBFPW~X-%L_XB0sz@zKwR zw>n5A{UFe*`^g}8=e~SA3IuBWXaG_zM@_I+R#yC8fdM!{5saK~vh8kT@7 zKm#u}fV6JFw4Fn9L_nR#0U-VgfHm+0NVfAnknas3p?`8gZ=x&#YMz>!TEu}s6aH`uBt3TPo9eO;rQB{~kF F{{v7JK?(o> literal 0 HcmV?d00001 diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index bfc68ab4..5d49375d 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -98,11 +98,11 @@ or equivalently, if and only if ### D-separation -in previous sections we used conditional dependencies. but in big graph, find dependencies with algebraic methods is so difficult and complex. in this section we present an algorithm called D-separation to find different conditional dependencies. +In previous sections we used conditional dependencies. But in a big graph, finding dependencies with algebraic methods is so difficult and complex. In this section we present an algorithm called D-separation to find different conditional dependencies. -outline: +Outline: -in this method, first we check conditional dependency for triples (3 nodes that connected) and specify several rules to find dependencies in triples. then we use these rules to find conditional dependencies in general cases that have big graph. in another word, in big graphs we find triples and apply some processing on these sub-graphs. +In this method, first we check conditional dependency for triples (3 nodes that connected) and specify several rules to find dependencies in triples. then we use these rules to find conditional dependencies in general cases that have big graph. in another word, in big graphs we find triples and apply some processing on these sub-graphs. in the following, we will examine the different situations for triples: #### Causal Chain @@ -136,7 +136,7 @@ is X is guaranteed to be independent of Z given Y? - the answer is YES. to prove this claim, we use algebraic operations. we should show equality of P(Z|Y) and P(Z|Y,X). -- $$P( z | x, y ) = \frac{P( x, y, z )}{P( x, y )} = \frac{P(x)P( y | x )P( z | y )}{P(x)P( y | x )} = P( z | y )$$ +![ch_formula1](assets/ch10.PNG) - we conclude that with condition on Y, X and Z are independent. @@ -175,7 +175,7 @@ is X is guaranteed to be independent of Z given Y? - the answer is Yes. to prove this claim, we use algebraic operations. we should show equality of P(Z|Y) and P(Z|Y,X). -- $$P( z | x, y ) = \frac{P( x, y, z )}{P( x, y )} = \frac{P(y)P( x | y )P( z | y )}{P(y)P( x | y )} = P( z | y )$$ +![ch_formula2](assets/ch11.PNG) - we conclude that with condition on Y, X and Z are independent. @@ -200,7 +200,7 @@ is X is guaranteed to be independent of Y? - the answer, unlike previous versions, is Yes. ballgame and raining do not get effect from each other or a common cause. to prove this claim, use algebraic operation again: -- $$P(X, Y) = \sum_{z \in Z} P(X, Y, z) = \sum_{z \in Z} P(X)P(Y)P(z|X, Y) = P(X)P(Y) \sum_{z \in Z} P(z|X, Y) = P(X)P(Y)$$ +![ch_formula1](assets/ch12.PNG) - So we show P(X,Y) = P(X)P(Y) and conclude X and Y are independent. @@ -248,29 +248,29 @@ Example: ![exp1](assets/ch5.jpg) - - **without evidence:** there is only one path between R and B that is a “common effect” triple. it is inactive so R and B are independent. - - **evidence=T:** there is a “common effect” triple that is active. So, independency of R and B aren’t guaranteed. - - **evidence=T’:** this situation is similar to previous evidence. +- **without evidence:** there is only one path between R and B that is a “common effect” triple. it is inactive so R and B are independent. +- **evidence=T:** there is a “common effect” triple that is active. So, independency of R and B aren’t guaranteed. +- **evidence=T’:** this situation is similar to previous evidence. - b. we want to check conditionally independency between L and B variables with different evidence. ![exp2](assets/ch6.jpg) - - **without evidence:** there is only one path between L and B that has two triples. L->R->T is active but R->T<-B is inactive. So, this pass is inactive. So, L and B are independent. - - **evidence = T:** similar to previous segment there is only one path that has two triples. both are active. So, L and B aren't guaranteed to be independent given T. - - **evidence = T':** it is similar to similar to segment. - - **evidence = T, R:** similar to 2’nd segment the R->T<-B triple is active but L->R->T is inactive. So, L and B are independent given T and R. +- **without evidence:** there is only one path between L and B that has two triples. L->R->T is active but R->T<-B is inactive. So, this pass is inactive. So, L and B are independent. +- **evidence = T:** similar to previous segment there is only one path that has two triples. both are active. So, L and B aren't guaranteed to be independent given T. +- **evidence = T':** it is similar to similar to segment. +- **evidence = T, R:** similar to 2’nd segment the R->T<-B triple is active but L->R->T is inactive. So, L and B are independent given T and R. - c. we want to check conditionally independency between T and D variables with different evidence. ![exp3](assets/ch7.jpg) - - **without evidence:** There are two paths between T and D. upper path is an active “common cause” and lower path is an inactive “common effect”. So, L and T aren't guaranteed to be independent. - - **evidence = R:** There are two paths between T and D. upper path is an inactive “common cause” and lower path is an inactive “common effect”. So, L and T are independent given R. - - **evidence = R, S:** There are two paths between T and D. upper path is an inactive “common cause” and lower path is an active “common effect”. So, L and T aren't guaranteed to be independent. +- **without evidence:** There are two paths between T and D. upper path is an active “common cause” and lower path is an inactive “common effect”. So, L and T aren't guaranteed to be independent. +- **evidence = R:** There are two paths between T and D. upper path is an inactive “common cause” and lower path is an inactive “common effect”. So, L and T are independent given R. +- **evidence = R, S:** There are two paths between T and D. upper path is an inactive “common cause” and lower path is an active “common effect”. So, L and T aren't guaranteed to be independent. ### Structure Implications ## Conclusion -## References +## References \ No newline at end of file From 755f42a60a8e3a83d74063a4bb09b26575b40997 Mon Sep 17 00:00:00 2001 From: GildedJson Date: Sat, 18 Dec 2021 00:25:32 -0800 Subject: [PATCH 09/59] Fixed grammer part1 --- notebooks/bayesian_networks_representation/index.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 5d49375d..9178a642 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -102,20 +102,20 @@ In previous sections we used conditional dependencies. But in a big graph, findi Outline: -In this method, first we check conditional dependency for triples (3 nodes that connected) and specify several rules to find dependencies in triples. then we use these rules to find conditional dependencies in general cases that have big graph. in another word, in big graphs we find triples and apply some processing on these sub-graphs. -in the following, we will examine the different situations for triples: +In this method, first we check conditional dependency for ***triples*** (3 nodes that are connected) and specify several rules to find whether the last and first variables are dependent or not. Then we use these rules to find conditional dependencies in general cases on bigger graph. In another words, in big graphs we do some processing on each triple in the path between the two desired nodes to find if they are independent. +In the following section, we will examine three different situations for triples: #### Causal Chain -first configuration is “casual chain” +The first configuration is “casual chain” -- in this configuration middle node transmits the effect of the previous node to the next one. +- In this configuration, middle node transmits the effect of the previous node to the next one. Example: ![causal_ex](assets/ch1.png) -in this BN, low pressure causes rain and rain causes traffic. in another word rain transmits the effect of low pressure to traffic. +In this BN, low pressure causes rain and rain causes traffic. in another word rain transmits the effect of low pressure to traffic. is X is guaranteed to be independent of Z? From 78ff261c1a9d1916ac116c3c241e777ae070cb4e Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 11:29:52 +0330 Subject: [PATCH 10/59] Update index.md add structure implication and topology limits distributions --- notebooks/bayesian_networks_representation/index.md | 12 +++++++++++- 1 file changed, 11 insertions(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 6c2476e2..b622f837 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -270,7 +270,17 @@ Example: - **evidence = R, S:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an active “common effect”. So, L and T aren't guaranteed to be independent. ### Structure Implications - +- As you can see, with the help of this algorithm, we can examine the conditional independence of two random variables. So, if we test the algorithm on all modes, we get a list of conditional independence. but sometimes this list isn’t complete. because when we check conditional dependencies, some cases aren’t certain, and using this method alone, their independence cannot be recognized. +-but in some cases, “d-separation” algorithm can find all dependencies. For example, in the figure below; + +![stIm](assets/sim.jpg) +### Topology Limits Distributions +-In a given graph topology, only certain joint distributions can be encoded. The graph structure guarantees certain (conditional) independence (There might be more independence). Adding arcs, increase the dependence of variables. +-For example, in the figure below, different kind of dependence for triples is mentioned. +![TLD](assets/tld.jpg) +-- Green color: triples in which every pair of RVs are independent. +-- Red color: triples in which two RVs are independent given the 3rd one. +-- Blue color: triples in which no independence is found. ## Conclusion ## References From a5affb237bc801cb0c498ec6d12e17f972a6f2e9 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 11:31:23 +0330 Subject: [PATCH 11/59] Add files via upload --- .../assets/sim.jpg | Bin 0 -> 27318 bytes .../assets/tld.jpg | Bin 0 -> 67504 bytes 2 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 notebooks/bayesian_networks_representation/assets/sim.jpg create mode 100644 notebooks/bayesian_networks_representation/assets/tld.jpg diff --git a/notebooks/bayesian_networks_representation/assets/sim.jpg 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ztjg?_hu{F5ZTH-tvUV{r6w=r^WlEcKQ#I|E-Xv#&Zz#+;%0R|>BSt{IBTHAvU_f&z zyuRrjgKuIjwR$hatf!e<>2N?rKk+c+`ZZ?n!xQtqd8>~P-xf;I%BfJh|8?cbY3-1} zM!l7TiO7utAspz4pZTa~KD~0ey9n&U>wo&s*xG-ND*mthUG(R7VUUn9p1vmheERQ~ zf`<8*#k+4i*nx&8uH#?Q+@9b2l3lO#0du}MT8LMJhNVJvDeZ|NGJ0 Fe*@J23kLuI literal 0 HcmV?d00001 From 4adb7fe94319c8aef9fb44940f08da656f075fb8 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 11:34:38 +0330 Subject: [PATCH 12/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index b622f837..cf07a756 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -275,12 +275,13 @@ Example: ![stIm](assets/sim.jpg) ### Topology Limits Distributions --In a given graph topology, only certain joint distributions can be encoded. The graph structure guarantees certain (conditional) independence (There might be more independence). Adding arcs, increase the dependence of variables. +- In a given graph topology, only certain joint distributions can be encoded. The graph structure guarantees certain (conditional) independence (There might be more independence). Adding arcs, increase the dependence of variables. -For example, in the figure below, different kind of dependence for triples is mentioned. ![TLD](assets/tld.jpg) --- Green color: triples in which every pair of RVs are independent. --- Red color: triples in which two RVs are independent given the 3rd one. --- Blue color: triples in which no independence is found. + +- Green color: triples in which every pair of RVs are independent. +- Red color: triples in which two RVs are independent given the 3rd one. +- Blue color: triples in which no independence is found. ## Conclusion ## References From e7391ce9c803815fa3d034fecd50166585941111 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 11:35:31 +0330 Subject: [PATCH 13/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index cf07a756..04da9403 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -265,9 +265,9 @@ Example: ![exp3](assets/ch7.jpg) -- **without evidence:** There are two paths between T and D. upper path is an active “common cause” and the lower path is an inactive “common effect”. So, L and T aren't guaranteed to be independent. -- **evidence = R:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an inactive “common effect”. So, L and T are independent given R. -- **evidence = R, S:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an active “common effect”. So, L and T aren't guaranteed to be independent. +> **without evidence:** There are two paths between T and D. upper path is an active “common cause” and the lower path is an inactive “common effect”. So, L and T aren't guaranteed to be independent. +> **evidence = R:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an inactive “common effect”. So, L and T are independent given R. +> **evidence = R, S:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an active “common effect”. So, L and T aren't guaranteed to be independent. ### Structure Implications - As you can see, with the help of this algorithm, we can examine the conditional independence of two random variables. So, if we test the algorithm on all modes, we get a list of conditional independence. but sometimes this list isn’t complete. because when we check conditional dependencies, some cases aren’t certain, and using this method alone, their independence cannot be recognized. From e619977f42cd8767bb7c3737baeac8650b6a35a4 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 11:36:20 +0330 Subject: [PATCH 14/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 04da9403..c786c7d8 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -261,13 +261,13 @@ Example: - **evidence = T':** it is similar to segment. - **evidence = T, R:** similar to the 2’nd segment the R->T<-B triple is active but L->R->T is inactive. So, L and B are independent given T and R. -- c. we want to check conditionally independence between T and D variables with different evidence. + c. we want to check conditionally independence between T and D variables with different evidence. ![exp3](assets/ch7.jpg) -> **without evidence:** There are two paths between T and D. upper path is an active “common cause” and the lower path is an inactive “common effect”. So, L and T aren't guaranteed to be independent. -> **evidence = R:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an inactive “common effect”. So, L and T are independent given R. -> **evidence = R, S:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an active “common effect”. So, L and T aren't guaranteed to be independent. +- **without evidence:** There are two paths between T and D. upper path is an active “common cause” and the lower path is an inactive “common effect”. So, L and T aren't guaranteed to be independent. +- **evidence = R:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an inactive “common effect”. So, L and T are independent given R. +- **evidence = R, S:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an active “common effect”. So, L and T aren't guaranteed to be independent. ### Structure Implications - As you can see, with the help of this algorithm, we can examine the conditional independence of two random variables. So, if we test the algorithm on all modes, we get a list of conditional independence. but sometimes this list isn’t complete. because when we check conditional dependencies, some cases aren’t certain, and using this method alone, their independence cannot be recognized. From e7dc5944222734ffa43ebcc4995024b03542f577 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 11:39:14 +0330 Subject: [PATCH 15/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index c786c7d8..f22d1745 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -20,6 +20,7 @@ - [Common Cause](#Common-Cause) - [V-Structure](#V-Structure) - [Reachability](#Reachability) + - [Topology Limits Distributions](#Topology-Limits-Distributions) - [Active/Inactive Paths](#Active/Inactive-Paths) - [Structure Implications](#Structure-Implications) - [Conclusion](#Conclusion) @@ -275,8 +276,8 @@ Example: ![stIm](assets/sim.jpg) ### Topology Limits Distributions -- In a given graph topology, only certain joint distributions can be encoded. The graph structure guarantees certain (conditional) independence (There might be more independence). Adding arcs, increase the dependence of variables. --For example, in the figure below, different kind of dependence for triples is mentioned. +In a given graph topology, only certain joint distributions can be encoded. The graph structure guarantees certain (conditional) independence (There might be more independence). Adding arcs, increase the dependence of variables. +For example, in the figure below, different kind of dependence for triples is mentioned. ![TLD](assets/tld.jpg) - Green color: triples in which every pair of RVs are independent. From 29bd89a79d9e641e7c2330c65e4d9535f42356de Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 11:45:09 +0330 Subject: [PATCH 16/59] Update index.md --- .../bayesian_networks_representation/index.md | 49 +++++++++---------- 1 file changed, 23 insertions(+), 26 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index f22d1745..f07027e6 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -20,9 +20,9 @@ - [Common Cause](#Common-Cause) - [V-Structure](#V-Structure) - [Reachability](#Reachability) - - [Topology Limits Distributions](#Topology-Limits-Distributions) - [Active/Inactive Paths](#Active/Inactive-Paths) - [Structure Implications](#Structure-Implications) + - [Topology Limits Distributions](#Topology-Limits-Distributions) - [Conclusion](#Conclusion) - [References](#References) @@ -151,12 +151,10 @@ second configuration is “common cause”: - in this configuration one variable effect on another two variables. but child nodes are not related directly. -- P(x, y, z) = P(y) P(x|y) P(z|y) - Example: ![common_cause_ex](assets/ch2.jpg) - +- P(x, y, z) = P(y) P(x|y) P(z|y)
in this BN, project due causes lab full and forums busy. is X is guaranteed to be independent of Z? @@ -185,49 +183,48 @@ is X is guaranteed to be independent of Z given Y? #### V-Structure -the last configuration is "common effect" and is sometimes called V-Structure. - -- in this configuration, one variable is affected by another two variables. +the last configuration is "common effect" and is sometimes called V-Structure.
-- P(x, y, z) = P(x) P(y) P(z|x, y) +in this configuration, one variable is affected by another two variables.
Example: ![common_effect_ex](assets/ch3.jpg) +- P(x, y, z) = P(x) P(y) P(z|x, y)
-in following BN, both ballgame and raining cause traffic. +in following BN, both ballgame and raining cause traffic.
-is X is guaranteed to be independent of Y? +is X is guaranteed to be independent of Y?
-- the answer, unlike previous versions, is Yes. ballgame and rain do not get effect from each other or a common cause. to prove this claim, use algebraic operation again: + the answer, unlike previous versions, is Yes. ballgame and rain do not get effect from each other or a common cause. to prove this claim, use algebraic operation again: ![ch_formula1](assets/ch12.PNG) - So we show P(X,Y) = P(X)P(Y) and conclude X and Y are independent. -is X is guaranteed to be independent of Y given z? +is X is guaranteed to be independent of Y given z?
-- the answer is NO. with an example we describe the situation. consider X and Y have random values and probability of occurrence each one is 0.5. (P(+x)=P(+y)=0.5) and traffic will occur when ballgame or raining occurs. + the answer is NO. with an example we describe the situation. consider X and Y have random values and probability of occurrence each one is 0.5. (P(+x)=P(+y)=0.5) and traffic will occur when ballgame or raining occurs. -- P(+z|+x,+y)=P(+z|+x,-y)=P(+z|-x,+y)=1 P(+z|-x,-y)=0 +- P(+z|+x,+y)=P(+z|+x,-y)=P(+z|-x,+y)=1 P(+z|-x,-y)=0
-- From the problem description, We can conclude that P(+x|+z,-y)=1 and P(+x|-y)=0.5. So, P(X|Z,Y) is not equal to P(X|Y). +From the problem description, We can conclude that P(+x|+z,-y)=1 and P(+x|-y)=0.5. So, P(X|Z,Y) is not equal to P(X|Y).
-- So, with the observation of Z, X and Y are not independent. this means Observing an effect, activating influence between possible causes. +So, with the observation of Z, X and Y are not independent. this means Observing an effect, activating influence between possible causes. ### Reachability -Now, we want to use these three cases to check conditional dependency in any arbitrary graph. generally, the problem is checking dependency between two random variables in a BN in which some (or any) variables are observed. for this goal, we break the graph into triples which we learned above, and do some checking on them. In the following, these steps are described. +Now, we want to use these three cases to check conditional dependency in any arbitrary graph. generally, the problem is checking dependency between two random variables in a BN in which some (or any) variables are observed. for this goal, we break the graph into triples which we learned above, and do some checking on them. In the following, these steps are described.
in the first step, we should shade evidence nodes, nodes that are observed in the problem. then looking for undirected paths between determined random variables. the first idea is that if two nodes are connected by a path that is blocked by a shaded node, two random variables are independent. But there are drawbacks to this rule. when we have several paths between two RVs or there is a V-structure triple that its bottom node is shaded, our method is wrong. So, we describe the next step to correct these problems. ### Active/Inactive Paths -Question: are X and Y conditionally independent given evidence variables {Z}? +Question: are X and Y conditionally independent given evidence variables {Z}?
-- Yes, if x and y “d-separated” by z. + Yes, if x and y “d-separated” by z. -to describe the "d-separated" concept, we first need to understand the concept of active/inactive path. +to describe the "d-separated" concept, we first need to understand the concept of active/inactive path.
a path is active if each triple of it is active. a triple is active if: @@ -235,7 +232,7 @@ a path is active if each triple of it is active. a triple is active if: - Common cause A <- B -> C where B is unobserved - Common effect (aka v-structure) A -> B <- C where B or one of its descendants is observed -Now we consider all undirected paths from X to Y after shading evidence variables, if none of these paths aren’t active paths, we can say X and Y are d-separated by Z. +Now we consider all undirected paths from X to Y after shading evidence variables, if none of these paths aren’t active paths, we can say X and Y are d-separated by Z.
in the figure below, examples of active and inactive triples are shown. @@ -245,7 +242,7 @@ note that a single inactive triple makes a path inactive. but two RVs are indepe Example: -- a. we want to check conditionally independence between R and B variables with different evidence. + a. we want to check conditionally independence between R and B variables with different evidence. ![exp1](assets/ch5.jpg) @@ -253,7 +250,7 @@ Example: - **evidence=T:** there is a “common effect” triple that is active. So, the independence of R and B isn’t guaranteed. - **evidence=T’:** this situation is similar to previous evidence. -- b. we want to check conditionally independence between L and B variables with different evidence. + b. we want to check conditionally independence between L and B variables with different evidence. ![exp2](assets/ch6.jpg) @@ -271,12 +268,12 @@ Example: - **evidence = R, S:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an active “common effect”. So, L and T aren't guaranteed to be independent. ### Structure Implications -- As you can see, with the help of this algorithm, we can examine the conditional independence of two random variables. So, if we test the algorithm on all modes, we get a list of conditional independence. but sometimes this list isn’t complete. because when we check conditional dependencies, some cases aren’t certain, and using this method alone, their independence cannot be recognized. --but in some cases, “d-separation” algorithm can find all dependencies. For example, in the figure below; + As you can see, with the help of this algorithm, we can examine the conditional independence of two random variables. So, if we test the algorithm on all modes, we get a list of conditional independence. but sometimes this list isn’t complete. because when we check conditional dependencies, some cases aren’t certain, and using this method alone, their independence cannot be recognized.
+but in some cases, “d-separation” algorithm can find all dependencies. For example, in the figure below; ![stIm](assets/sim.jpg) ### Topology Limits Distributions -In a given graph topology, only certain joint distributions can be encoded. The graph structure guarantees certain (conditional) independence (There might be more independence). Adding arcs, increase the dependence of variables. +In a given graph topology, only certain joint distributions can be encoded. The graph structure guarantees certain (conditional) independence (There might be more independence). Adding arcs, increase the dependence of variables.
For example, in the figure below, different kind of dependence for triples is mentioned. ![TLD](assets/tld.jpg) From 8c23e3a0787cce2a93d3d498a7f9a3f5996055ef Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 11:54:36 +0330 Subject: [PATCH 17/59] Update index.md --- .../bayesian_networks_representation/index.md | 77 ++++++++++--------- 1 file changed, 39 insertions(+), 38 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index f07027e6..88631ea8 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -16,6 +16,7 @@ - [Independence in Bayes' Nets](#Independence-in-Bayes'-Nets) - [Independency Assumptions](#Independency-Assumptions) - [D-separation](#D-separation) + - [Outline] (#Outline) - [Causal Chain](#Causal-Chain) - [Common Cause](#Common-Cause) - [V-Structure](#V-Structure) @@ -101,85 +102,85 @@ or equivalently, if and only if In previous sections we used conditional dependencies. But in a big graph, finding dependencies with algebraic methods is so difficult and complex. In this section we present an algorithm called D-separation to find different conditional dependencies. -Outline: +#### Outline: -In this method, first we check conditional dependency for ***triples*** (3 nodes that are connected) and specify several rules to find whether the last and first variables are dependent or not. Then we use these rules to find conditional dependencies in general cases on bigger graph. In another words, in big graphs we do some processing on each triple in the path between the two desired nodes to find if they are independent. +In this method, first we check conditional dependency for ***triples*** (3 nodes that are connected) and specify several rules to find whether the last and first variables are dependent or not. Then we use these rules to find conditional dependencies in general cases on bigger graph. In another words, in big graphs we do some processing on each triple in the path between the two desired nodes to find if they are independent.
In the following section, we will examine three different situations for triples: #### Causal Chain -The first configuration is “casual chain” +The first configuration is “casual chain”.
-- In this configuration, middle node transmits the effect of the previous node to the next one. +In this configuration, middle node transmits the effect of the previous node to the next one. Example: ![causal_ex](assets/ch1.png) -In this BN, low pressure causes rain and rain causes traffic. in another word rain transmits the effect of low pressure to traffic. +In this BN, low pressure causes rain and rain causes traffic. in another word rain transmits the effect of low pressure to traffic.
-is X is guaranteed to be independent of Z? +is X is guaranteed to be independent of Z?
-- the answer is NO. To prove this claim, it is enough to give an example that X and Z are dependent. suppose that X is random, and probability of low pressure (P(+x)) and high pressure(P(-x)) are equal to 0.5. P(+x)=P(-x)=0.5 +the answer is NO. To prove this claim, it is enough to give an example that X and Z are dependent. suppose that X is random, and probability of low pressure (P(+x)) and high pressure(P(-x)) are equal to 0.5. P(+x)=P(-x)=0.5
-- rainfall is completely dependent on low pressure and if low pressure occurs, rainfall is definite. and if high pressure occurs, rainfall is impossible. -P(+y|-x)=0 P(-y|-x)=1 P(+y|+x)=1 P(-y|+x)=0 +rainfall is completely dependent on low pressure and if low pressure occurs, rainfall is definite. and if high pressure occurs, rainfall is impossible.
+P(+y|-x)=0 P(+y|+x)=1
-- Also, traffic is completely dependent on rainfall. if it rains, traffic will occur and otherwise there isn’t any traffic. +Also, traffic is completely dependent on rainfall. if it rains, traffic will occur and otherwise there isn’t any traffic.
-- P(+z|+x)=1 P(-z|+x)= 0 P(+z|-x)= 0 P(-z|-x)= 1 +P(+z|+x)=1 P(-z|+x)= 0 P(+z|-x)= 0 P(-z|-x)= 1
-- now we should compare P(Z) and P(Z|X) to find dependency. for example low pressure and traffic occurred. (X=+x and Z=+z) we know P(+z|+x)=1 but P(+z)=0.5 +now we should compare P(Z) and P(Z|X) to find dependency. for example low pressure and traffic occurred. (X=+x and Z=+z) we know P(+z|+x)=1 but P(+z)=0.5
-- As a result, the two variables are not independent. +As a result, the two variables are not independent.
-is X is guaranteed to be independent of Z given Y? +is X is guaranteed to be independent of Z given Y?
-- the answer is YES. to prove this claim, we use algebraic operations. we should show equality of P(Z|Y) and P(Z|Y,X). +the answer is YES. to prove this claim, we use algebraic operations. we should show equality of P(Z|Y) and P(Z|Y,X). ![ch_formula1](assets/ch10.PNG) -- we conclude that with condition on Y, X and Z are independent. +we conclude that with condition on Y, X and Z are independent.
-- So, in triple casual case, if middle node is observed, end nodes are independent. because the effect of X to Z is not direct, if value of Y be known, change in X’ value causes no effect on Z’ value. in another word, if we have value of Y, value of X does not give any additional information about Z. +So, in triple casual case, if middle node is observed, end nodes are independent. because the effect of X to Z is not direct, if value of Y be known, change in X’ value causes no effect on Z’ value. in another word, if we have value of Y, value of X does not give any additional information about Z.
-- So, evidence along the chain “blocks” the influence. +So, evidence along the chain “blocks” the influence. #### Common Cause -second configuration is “common cause”: +second configuration is “common cause”:
-- in this configuration one variable effect on another two variables. but child nodes are not related directly. +in this configuration one variable effect on another two variables. but child nodes are not related directly.
Example: ![common_cause_ex](assets/ch2.jpg) - P(x, y, z) = P(y) P(x|y) P(z|y)
-in this BN, project due causes lab full and forums busy. +in this BN, project due causes lab full and forums busy.
-is X is guaranteed to be independent of Z? +is X is guaranteed to be independent of Z?
-- the answer is NO. it is enough to give an example set of CPTs that X and Z are dependent. suppose that Y is random, and the probability of being close to project deadline (P(+y)) and otherwise (P(-y)) are equal to 0.5. P(+y)=P(-y)=0.5 +the answer is NO. it is enough to give an example set of CPTs that X and Z are dependent. suppose that Y is random, and the probability of being close to project deadline (P(+y)) and otherwise (P(-y)) are equal to 0.5. P(+y)=P(-y)=0.5
-- laboratory fullness is completely dependent on project deadline and if deadline is near, laboratory is full. and if project deadline is not near, laboratory is not full. -P(+z|+y)=1 P(+z|-y)=0 +laboratory fullness is completely dependent on project deadline and if deadline is near, laboratory is full. and if project deadline is not near, laboratory is not full.
+P(+z|+y)=1 P(+z|-y)=0
-- Also, forums are related to project deadline. if deadline is near, forums get busy. -P(+x|+y)=1 P(+x|-y)=0 +Also, forums are related to project deadline. if deadline is near, forums get busy.
+P(+x|+y)=1 P(+x|-y)=0
-- now we should compare P(Z) and P(Z|X) to find dependency. for example, lab is full and forums are busy. (X=+x and Z=+z) we know P(+z|+x)=1 but P(+z)=0.5 -As a result, the two child variables in common cause triples are not independent in general case. +now we should compare P(Z) and P(Z|X) to find dependency. for example, lab is full and forums are busy. (X=+x and Z=+z) we know P(+z|+x)=1 but P(+z)=0.5 +As a result, the two child variables in common cause triples are not independent in general case.
-is X is guaranteed to be independent of Z given Y? +is X is guaranteed to be independent of Z given Y?
-- the answer is Yes. to prove this claim, we use algebraic operations. we should show equality of P(Z|Y) and P(Z|Y,X). +the answer is Yes. to prove this claim, we use algebraic operations. we should show equality of P(Z|Y) and P(Z|Y,X). ![ch_formula2](assets/ch11.PNG) -- we conclude that with condition on Y, X and Z are independent. +we conclude that with condition on Y, X and Z are independent.
-- So, in triple common cause, if parent node is observed, child nodes are independent. because the effect of X to Z is not direct, if value of Y be known, change in X’ value causes no effect on Z’ value. in another word, if we have value of Y, value of X does not give any additional information about Z. -- So, observing the cause blocks influence between effects. +So, in triple common cause, if parent node is observed, child nodes are independent. because the effect of X to Z is not direct, if value of Y be known, change in X’ value causes no effect on Z’ value. in another word, if we have value of Y, value of X does not give any additional information about Z.
+So, observing the cause blocks influence between effects. #### V-Structure @@ -196,15 +197,15 @@ in following BN, both ballgame and raining cause traffic.
is X is guaranteed to be independent of Y?
- the answer, unlike previous versions, is Yes. ballgame and rain do not get effect from each other or a common cause. to prove this claim, use algebraic operation again: +the answer, unlike previous versions, is Yes. ballgame and rain do not get effect from each other or a common cause. to prove this claim, use algebraic operation again: ![ch_formula1](assets/ch12.PNG) -- So we show P(X,Y) = P(X)P(Y) and conclude X and Y are independent. +So we show P(X,Y) = P(X)P(Y) and conclude X and Y are independent.
is X is guaranteed to be independent of Y given z?
- the answer is NO. with an example we describe the situation. consider X and Y have random values and probability of occurrence each one is 0.5. (P(+x)=P(+y)=0.5) and traffic will occur when ballgame or raining occurs. +the answer is NO. with an example we describe the situation. consider X and Y have random values and probability of occurrence each one is 0.5. (P(+x)=P(+y)=0.5) and traffic will occur when ballgame or raining occurs. - P(+z|+x,+y)=P(+z|+x,-y)=P(+z|-x,+y)=1 P(+z|-x,-y)=0
@@ -222,7 +223,7 @@ in the first step, we should shade evidence nodes, nodes that are observed in th Question: are X and Y conditionally independent given evidence variables {Z}?
- Yes, if x and y “d-separated” by z. +Yes, if x and y “d-separated” by z. to describe the "d-separated" concept, we first need to understand the concept of active/inactive path.
From 6421b5baa1adf44b82fb16edf026d7ad4ca663f7 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 11:55:09 +0330 Subject: [PATCH 18/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 88631ea8..8480c0b3 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -16,7 +16,7 @@ - [Independence in Bayes' Nets](#Independence-in-Bayes'-Nets) - [Independency Assumptions](#Independency-Assumptions) - [D-separation](#D-separation) - - [Outline] (#Outline) + - [Outline](#Outline) - [Causal Chain](#Causal-Chain) - [Common Cause](#Common-Cause) - [V-Structure](#V-Structure) From eeaba654192723124eacc3ec6e650d800d13b1cf Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 11:56:34 +0330 Subject: [PATCH 19/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 1 + 1 file changed, 1 insertion(+) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 8480c0b3..0a97a9a6 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -156,6 +156,7 @@ Example: ![common_cause_ex](assets/ch2.jpg) - P(x, y, z) = P(y) P(x|y) P(z|y)
+ in this BN, project due causes lab full and forums busy.
is X is guaranteed to be independent of Z?
From 5bfcb6b1545cf80215c6f409982c89fab9a9b8ce Mon Sep 17 00:00:00 2001 From: GildedJson Date: Sat, 18 Dec 2021 17:36:50 -0800 Subject: [PATCH 20/59] My last commit --- .../bayesian_networks_representation/index.md | 20 +++++++++++++++++++ 1 file changed, 20 insertions(+) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 0a97a9a6..a031e41a 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -25,6 +25,7 @@ - [Structure Implications](#Structure-Implications) - [Topology Limits Distributions](#Topology-Limits-Distributions) - [Conclusion](#Conclusion) + - [What comes next](#What-comes-next) - [References](#References) ## Intoduction @@ -270,18 +271,37 @@ Example: - **evidence = R, S:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an active “common effect”. So, L and T aren't guaranteed to be independent. ### Structure Implications + As you can see, with the help of this algorithm, we can examine the conditional independence of two random variables. So, if we test the algorithm on all modes, we get a list of conditional independence. but sometimes this list isn’t complete. because when we check conditional dependencies, some cases aren’t certain, and using this method alone, their independence cannot be recognized.
but in some cases, “d-separation” algorithm can find all dependencies. For example, in the figure below; ![stIm](assets/sim.jpg) + ### Topology Limits Distributions + In a given graph topology, only certain joint distributions can be encoded. The graph structure guarantees certain (conditional) independence (There might be more independence). Adding arcs, increase the dependence of variables.
For example, in the figure below, different kind of dependence for triples is mentioned. + ![TLD](assets/tld.jpg) - Green color: triples in which every pair of RVs are independent. - Red color: triples in which two RVs are independent given the 3rd one. - Blue color: triples in which no independence is found. + ## Conclusion +Through this lecture note, we saw that: + +- Joint distributions can be encoded efficiently using bayes' nets which simplifies calculations a lot compared to the classic joint distribution tables method. + +- Guaranteed independencies of distributions can be deducted from the bayesian net graph structure which weren't apparent at first. Moreover, using d-separation gives us precise conditional independence guarantees from the graph alone. + +- Even that isn't all. There may as well be more (conditional) independencies that are not detectable until we inspect the specific distributions. + +### What comes next + +Up until now, we saw how to build a bayes' net and how to find the independencies. The next step is to put these bayes' nets to use and infer probabilities from it. It's basically done by multiplying probabilities taken from every node on the path from top to the desired node. Read the next lecture note to learn all the details. + ## References + + \ No newline at end of file From ccb4993c8c479490b7c92260356a3346bfb53e77 Mon Sep 17 00:00:00 2001 From: GildedJson Date: Sat, 18 Dec 2021 19:56:26 -0800 Subject: [PATCH 21/59] Added chain rule --- .../assets/T2_Formula7.PNG | Bin 0 -> 9243 bytes .../bayesian_networks_representation/index.md | 14 ++++++++++++++ 2 files changed, 14 insertions(+) create mode 100644 notebooks/bayesian_networks_representation/assets/T2_Formula7.PNG diff --git a/notebooks/bayesian_networks_representation/assets/T2_Formula7.PNG b/notebooks/bayesian_networks_representation/assets/T2_Formula7.PNG new file mode 100644 index 0000000000000000000000000000000000000000..40835c58a43108685c17d2de6633a00cb2b585d0 GIT binary patch literal 9243 zcmcJ#Wl$VZ*DXv4?vvmy3Bfh^0KpO@1b1hEArRc%-7Po-4<6hG4-67)g1fsj=s>vh zeD%EFt^4Etyt}IVobKw==j^ptuf41KM5%uHgpWgsgMxyBuOKh0j)H>PiR^P@VIa?^ zB+hNf=7o#;Cux+Laq0u)1sYgNSqcTEE&=z^3>|rm?I^G3f`UT&_CLo91@$-QC@4(7 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vWX<9Kd!vd)tla;@4K4rooip);caq&Wm>9z&&~D_09ux(+FS0e#rosOU^LbPY literal 0 HcmV?d00001 diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index a031e41a..00a88801 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -79,6 +79,20 @@ or equivalently, if and only if ### Chain Rule +We all know the basic form of chain rule: + +![formula7](assets/T2_Formula7.PNG) + +Imagine we want P(X, Y, Z). Using the chain rule, we can write: + +- P(X,Y,Z) = P(X)P(Y|X)P(Z|X,Y) + +But if we know that Z and Y are independent, we can instead write: + +- P(X,Y,Z) = P(X)P(Y|X)P(Z|X) + +These independencies can be derived from bayes' nets. This will come in handy when we want to infer probabilities from bayes nets. + ## Bayes' Nets ![fridge](assets/Fridge_net.PNG) From dd1d2e92e29c636b7ca32a39ff249b89da94ece2 Mon Sep 17 00:00:00 2001 From: MohammadHosein Gheisarieh <44923177+mhgheisarieh@users.noreply.github.com> Date: Sat, 18 Dec 2021 21:27:16 +0330 Subject: [PATCH 22/59] Update index.md Complete Problems with joint distribution tables and Graphical Notation --- .../bayesian_networks_representation/index.md | 15 ++++++++++++++- 1 file changed, 14 insertions(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 00a88801..f3434474 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -99,8 +99,20 @@ These independencies can be derived from bayes' nets. This will come in handy wh ### Problems with joint distribution tables +There are two problems with using full joint distribution tables as our probabilistic models: +1. Unless there are only a few variables, the joint is WAY too big to represent explicitly. +2. Hard to learn (estimate) anything empirically about more than a few variables at a time. + +Bayes nets is a technique for describing complex joint distributions or models using simple and local distributions. + ### Graphical Notation +- There are some nodes related to variables (with domains). This nodes can be assigned (observed) or unassigned (unobserved) +- There are some arcs equales to interactions. This nodes are similar to CSP constraints and indicate direct influence between variables. And there are formally, it means the ares encode conditional independence. + +![image](https://user-images.githubusercontent.com/44923177/146651162-96c91d1f-9f38-4825-9d6f-983f36aca273.png) + + ### Semantics ### Probabilities in Bayes' Nets @@ -318,4 +330,5 @@ Up until now, we saw how to build a bayes' net and how to find the independencie ## References - \ No newline at end of file + + From e5c867f277a1605483240eb9fe4744c6d932fa0f Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 21:42:12 +0330 Subject: [PATCH 23/59] Create index.md --- .../bayesian_networks_representation/index.md | 52 ++++++++++++++----- 1 file changed, 38 insertions(+), 14 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index f3434474..4765306a 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -99,27 +99,52 @@ These independencies can be derived from bayes' nets. This will come in handy wh ### Problems with joint distribution tables -There are two problems with using full joint distribution tables as our probabilistic models: -1. Unless there are only a few variables, the joint is WAY too big to represent explicitly. -2. Hard to learn (estimate) anything empirically about more than a few variables at a time. - -Bayes nets is a technique for describing complex joint distributions or models using simple and local distributions. - ### Graphical Notation -- There are some nodes related to variables (with domains). This nodes can be assigned (observed) or unassigned (unobserved) -- There are some arcs equales to interactions. This nodes are similar to CSP constraints and indicate direct influence between variables. And there are formally, it means the ares encode conditional independence. - -![image](https://user-images.githubusercontent.com/44923177/146651162-96c91d1f-9f38-4825-9d6f-983f36aca273.png) - - ### Semantics ### Probabilities in Bayes' Nets + in BNs, join distribution is obtained by the product of local conditional distributions. So, to find the probability of a full assignment, multiply all relevant conditional probabilities. +![joint](assets/joint.PNG) + in the following, we will prove the correctness of this method.
+ Note first the chain rule that applies to each distribution: +![chain](assets/chain.PNG) + Now with respect to conditional independence, we can claim that: +![conditionaldep](assets/conditionaldep.PNG) + because xi is conditional independent of other nodes, given its parents.
+ So, we can conclude that: +![joint](assets/joint.PNG) + Note that not every BN can produce every distribution, but the BN topology determines what conditional independence can be produced.
+ + The following are some examples of BN distribution.
+ + a. coin flips
+ In this case, a coin is tossed n times, the probability of a head or tail being equal in each toss. If these actions are independent of each other, then it is as follows. +![coin](assets/coin.PNG) + for example P(H, T, T, H) = P(H)P(T)P(T)P(H)
+ Note that only distributions whose variables are absolutely independent can be represented by a Bayes’ net with no arcs.
+ + b. traffic
+ In this example, R stands for rain and T stands for traffic. It is also considered that rain causes traffic. +![traffic](assets/traffic.PNG) + for example P(+r, -t) = P(+r)P(-t|+r)
+ + c. alarm network
+ In this case, it is assumed that the house alarm is sounded by an earthquake or burglary. John and Mary may also call us if the alarm sounds. + ![alarm](assets/alarm.PNG) + +For example P(+b,-e,+a,-j,-m)=P(+b)P(-e)P(+a|+b,-e)P(-j|+a)P(-m|+a) ### Causality in Bayes' Nets - + If in a Bayes' net the arrows show the real causes, it can be better investigated and the probabilities found. The network also becomes simpler. Because the number of parents is less.
+ Sometimes arrows do not show causal relationships. Or even in a network, none of the arrows have a causal relationship. For example, when some important variables are not available in the network. In such cases, the arrows show correlation and not causation.
+ + Therefore, we do not need to know the exact causal structure between the variables. ### Space Efficiency + The CPT size of a node if it has m different value and has k parents, each of which also has m different value, is equal to m^(k+1).
+ Therefore, in a Bayes' net with N Boolean variables, if each node has a maximum of k parents, the size of the whole net is of the order of O(N*2^(k+1)). Whereas if we want to store their joint distribution, it is the size of 2^N.
+ + So, we conclude that if we use this method, it usually requires less memory. It is also easier and faster to use local CPTs. ## Independence in Bayes' Nets @@ -331,4 +356,3 @@ Up until now, we saw how to build a bayes' net and how to find the independencie ## References - From 3118d5ac84c4a5a5803143fb02e8eb7ce0e4700b Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 21:44:02 +0330 Subject: [PATCH 24/59] Add files via upload --- .../assets/alarm.png | Bin 0 -> 70469 bytes .../assets/chain.png | Bin 0 -> 4782 bytes .../assets/coin.png | Bin 0 -> 56615 bytes .../assets/conditionaldep.png | Bin 0 -> 4081 bytes .../assets/joint.png | Bin 0 -> 5051 bytes .../assets/traffic.png | Bin 0 -> 19338 bytes 6 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 notebooks/bayesian_networks_representation/assets/alarm.png create mode 100644 notebooks/bayesian_networks_representation/assets/chain.png create mode 100644 notebooks/bayesian_networks_representation/assets/coin.png create mode 100644 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Note first the chain rule that applies to each distribution: -![chain](assets/chain.PNG) +![chain](assets/chain.png) Now with respect to conditional independence, we can claim that: ![conditionaldep](assets/conditionaldep.PNG) because xi is conditional independent of other nodes, given its parents.
So, we can conclude that: -![joint](assets/joint.PNG) +![joint](assets/joint.png) Note that not every BN can produce every distribution, but the BN topology determines what conditional independence can be produced.
The following are some examples of BN distribution.
a. coin flips
In this case, a coin is tossed n times, the probability of a head or tail being equal in each toss. If these actions are independent of each other, then it is as follows. -![coin](assets/coin.PNG) +![coin](assets/coin.png) for example P(H, T, T, H) = P(H)P(T)P(T)P(H)
Note that only distributions whose variables are absolutely independent can be represented by a Bayes’ net with no arcs.
b. traffic
In this example, R stands for rain and T stands for traffic. It is also considered that rain causes traffic. -![traffic](assets/traffic.PNG) +![traffic](assets/traffic.png) for example P(+r, -t) = P(+r)P(-t|+r)
c. alarm network
In this case, it is assumed that the house alarm is sounded by an earthquake or burglary. John and Mary may also call us if the alarm sounds. - ![alarm](assets/alarm.PNG) + ![alarm](assets/alarm.png) For example P(+b,-e,+a,-j,-m)=P(+b)P(-e)P(+a|+b,-e)P(-j|+a)P(-m|+a) From bc9d539f109f846a74c01cbe79fecd40bb035899 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 21:49:43 +0330 Subject: [PATCH 26/59] Update index.md --- .../bayesian_networks_representation/index.md | 15 +++++++-------- 1 file changed, 7 insertions(+), 8 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 69a026fb..0744dfbb 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -105,34 +105,33 @@ These independencies can be derived from bayes' nets. This will come in handy wh ### Probabilities in Bayes' Nets in BNs, join distribution is obtained by the product of local conditional distributions. So, to find the probability of a full assignment, multiply all relevant conditional probabilities. -![joint](assets/joint.png) +![joint](assets/joint.png)
in the following, we will prove the correctness of this method.
Note first the chain rule that applies to each distribution: -![chain](assets/chain.png) +![chain](assets/chain.png)
Now with respect to conditional independence, we can claim that: -![conditionaldep](assets/conditionaldep.PNG) +![conditionaldep](assets/conditionaldep.png)
because xi is conditional independent of other nodes, given its parents.
So, we can conclude that: -![joint](assets/joint.png) +![joint](assets/joint.png)
Note that not every BN can produce every distribution, but the BN topology determines what conditional independence can be produced.
The following are some examples of BN distribution.
a. coin flips
In this case, a coin is tossed n times, the probability of a head or tail being equal in each toss. If these actions are independent of each other, then it is as follows. -![coin](assets/coin.png) +![coin](assets/coin.png)
for example P(H, T, T, H) = P(H)P(T)P(T)P(H)
Note that only distributions whose variables are absolutely independent can be represented by a Bayes’ net with no arcs.
b. traffic
In this example, R stands for rain and T stands for traffic. It is also considered that rain causes traffic. -![traffic](assets/traffic.png) +![traffic](assets/traffic.png)
for example P(+r, -t) = P(+r)P(-t|+r)
c. alarm network
In this case, it is assumed that the house alarm is sounded by an earthquake or burglary. John and Mary may also call us if the alarm sounds. - ![alarm](assets/alarm.png) - + ![alarm](assets/alarm.png)
For example P(+b,-e,+a,-j,-m)=P(+b)P(-e)P(+a|+b,-e)P(-j|+a)P(-m|+a) ### Causality in Bayes' Nets From e5baaa8bc13a332910f2a18e45b082a051ba1f3f Mon Sep 17 00:00:00 2001 From: MohammadHosein Gheisarieh <44923177+mhgheisarieh@users.noreply.github.com> Date: Sat, 18 Dec 2021 22:15:35 +0330 Subject: [PATCH 27/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 12 ++++++++++++ 1 file changed, 12 insertions(+) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 0744dfbb..98253e10 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -99,8 +99,19 @@ These independencies can be derived from bayes' nets. This will come in handy wh ### Problems with joint distribution tables +There are two problems with using full joint distribution tables as our probabilistic models: +1. Unless there are only a few variables, the joint is WAY too big to represent explicitly. +2. Hard to learn (estimate) anything empirically about more than a few variables at a time. + +Bayes nets is a technique for describing complex joint distributions or models using simple and local distributions. + ### Graphical Notation +- There are some nodes related to variables (with domains). This nodes can be assigned (observed) or unassigned (unobserved) +- There are some arcs equales to interactions. This nodes are similar to CSP constraints and indicate direct influence between variables. And there are formally, it means the ares encode conditional independence. + +![image](https://user-images.githubusercontent.com/44923177/146651162-96c91d1f-9f38-4825-9d6f-983f36aca273.png) + ### Semantics ### Probabilities in Bayes' Nets @@ -355,3 +366,4 @@ Up until now, we saw how to build a bayes' net and how to find the independencie ## References + From c7682135420f3a3bb4dc461468c17f1d23fe536a Mon Sep 17 00:00:00 2001 From: MohammadHosein Gheisarieh <44923177+mhgheisarieh@users.noreply.github.com> Date: Sat, 18 Dec 2021 22:23:51 +0330 Subject: [PATCH 28/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 98253e10..ecc64232 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -365,5 +365,5 @@ Up until now, we saw how to build a bayes' net and how to find the independencie ## References - +
From c9561caf7dd18c0e8fa6368014b813d91183626a Mon Sep 17 00:00:00 2001 From: GildedJson <60608967+GildedJson@users.noreply.github.com> Date: Sat, 18 Dec 2021 11:34:07 -0800 Subject: [PATCH 29/59] Some grammar fixing and deleted assumption --- .../bayesian_networks_representation/index.md | 21 ++++++++----------- 1 file changed, 9 insertions(+), 12 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index ecc64232..24cf4b1f 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -14,7 +14,6 @@ - [Causality in Bayes' Nets](#Causality-in-Bayes'-Nets) - [Space Efficiency](#Space-Efficiency) - [Independence in Bayes' Nets](#Independence-in-Bayes'-Nets) - - [Independency Assumptions](#Independency-Assumptions) - [D-separation](#D-separation) - [Outline](#Outline) - [Causal Chain](#Causal-Chain) @@ -117,31 +116,31 @@ Bayes nets is a technique for describing complex joint distributions or models u ### Probabilities in Bayes' Nets in BNs, join distribution is obtained by the product of local conditional distributions. So, to find the probability of a full assignment, multiply all relevant conditional probabilities. ![joint](assets/joint.png)
- in the following, we will prove the correctness of this method.
- Note first the chain rule that applies to each distribution: + in the following part, we will prove this method is correct.
+ First note the chain rule applied to each distribution: ![chain](assets/chain.png)
Now with respect to conditional independence, we can claim that: ![conditionaldep](assets/conditionaldep.png)
- because xi is conditional independent of other nodes, given its parents.
+ Which is true because xi is conditionally independent of other nodes, given its parents.
So, we can conclude that: ![joint](assets/joint.png)
Note that not every BN can produce every distribution, but the BN topology determines what conditional independence can be produced.
- The following are some examples of BN distribution.
+ See the following examples of BN distribution.
a. coin flips
- In this case, a coin is tossed n times, the probability of a head or tail being equal in each toss. If these actions are independent of each other, then it is as follows. + In this case, a coin is tossed n times, the probability of a head or tail being equal in each toss. If these actions are independent of each other, then the graph is as shown in the image. ![coin](assets/coin.png)
- for example P(H, T, T, H) = P(H)P(T)P(T)P(H)
+ For example P(H, T, T, H) = P(H)P(T)P(T)P(H)
Note that only distributions whose variables are absolutely independent can be represented by a Bayes’ net with no arcs.
b. traffic
- In this example, R stands for rain and T stands for traffic. It is also considered that rain causes traffic. + In this example, R stands for rain and T stands for traffic. It is also assumed that rain causes traffic. ![traffic](assets/traffic.png)
- for example P(+r, -t) = P(+r)P(-t|+r)
+ For example P(+r, -t) = P(+r)P(-t|+r)
c. alarm network
- In this case, it is assumed that the house alarm is sounded by an earthquake or burglary. John and Mary may also call us if the alarm sounds. + In this case, it is assumed that the house alarm goes off by an earthquake or burglary. John and Mary may also call us if the alarm is ringing. ![alarm](assets/alarm.png)
For example P(+b,-e,+a,-j,-m)=P(+b)P(-e)P(+a|+b,-e)P(-j|+a)P(-m|+a) @@ -158,8 +157,6 @@ For example P(+b,-e,+a,-j,-m)=P(+b)P(-e)P(+a|+b,-e)P(-j|+a)P(-m|+a) ## Independence in Bayes' Nets -### Independency Assumptions - ### D-separation In previous sections we used conditional dependencies. But in a big graph, finding dependencies with algebraic methods is so difficult and complex. In this section we present an algorithm called D-separation to find different conditional dependencies. From 9939f2b1e157457dc9785779f04bc6a0f561a371 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 23:11:55 +0330 Subject: [PATCH 30/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 24cf4b1f..b11bdb94 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -175,6 +175,7 @@ In this configuration, middle node transmits the effect of the previous node to Example: ![causal_ex](assets/ch1.png) +- P(X, Y, Z) = P(X) P(Y|X) P(Z|Y)
In this BN, low pressure causes rain and rain causes traffic. in another word rain transmits the effect of low pressure to traffic.
@@ -301,7 +302,7 @@ in the figure below, examples of active and inactive triples are shown. note that a single inactive triple makes a path inactive. but two RVs are independent if all paths between related nodes are inactive. So, if some paths were active, independence is not guaranteed. -Example: +Examples: a. we want to check conditionally independence between R and B variables with different evidence. From 703dba51736e97052cb2015185e6e28ec3e7bf66 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 23:13:47 +0330 Subject: [PATCH 31/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index b11bdb94..4b880743 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -215,7 +215,7 @@ in this configuration one variable effect on another two variables. but child no Example: ![common_cause_ex](assets/ch2.jpg) -- P(x, y, z) = P(y) P(x|y) P(z|y)
+- P(X, Y, Z) = P(Y) P(X|Y) P(Z|Y)
in this BN, project due causes lab full and forums busy.
@@ -252,7 +252,7 @@ in this configuration, one variable is affected by another two variables.
Example: ![common_effect_ex](assets/ch3.jpg) -- P(x, y, z) = P(x) P(y) P(z|x, y)
+- P(X, Y, Z) = P(X) P(Y) P(Z|X, Y)
in following BN, both ballgame and raining cause traffic.
From 94f9810d7d7c69517705e8bba1c7dc48118356a5 Mon Sep 17 00:00:00 2001 From: MohammadHosein Gheisarieh <44923177+mhgheisarieh@users.noreply.github.com> Date: Sat, 18 Dec 2021 23:46:48 +0330 Subject: [PATCH 32/59] add Semantics --- notebooks/bayesian_networks_representation/index.md | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 4b880743..040680bb 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -113,6 +113,10 @@ Bayes nets is a technique for describing complex joint distributions or models u ### Semantics +![image](https://user-images.githubusercontent.com/44923177/146654189-38c25839-e446-420f-af85-473900274dfb.png) + +In this picture there are a set of nodes, one per variable X. And we have a directed and acyclic graph. Each node of this graph shows a conditional distribution. + ### Probabilities in Bayes' Nets in BNs, join distribution is obtained by the product of local conditional distributions. So, to find the probability of a full assignment, multiply all relevant conditional probabilities. ![joint](assets/joint.png)
From 16347494d9da8ab94d06b3137839a0c184424c52 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 18 Dec 2021 23:50:47 +0330 Subject: [PATCH 33/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 040680bb..ef14f7bd 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -192,7 +192,7 @@ P(+y|-x)=0 P(+y|+x)=1
Also, traffic is completely dependent on rainfall. if it rains, traffic will occur and otherwise there isn’t any traffic.
-P(+z|+x)=1 P(-z|+x)= 0 P(+z|-x)= 0 P(-z|-x)= 1
+P(+z|+x)=1 P(-z|-x)= 1
now we should compare P(Z) and P(Z|X) to find dependency. for example low pressure and traffic occurred. (X=+x and Z=+z) we know P(+z|+x)=1 but P(+z)=0.5
From 262bc1b144d765356a3f97a3a1d5f5b91febd6e8 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Thu, 30 Dec 2021 02:08:12 +0330 Subject: [PATCH 34/59] Update index.md --- .../bayesian_networks_representation/index.md | 21 ++++++++++--------- 1 file changed, 11 insertions(+), 10 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index ef14f7bd..57b9bbb6 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -118,14 +118,14 @@ Bayes nets is a technique for describing complex joint distributions or models u In this picture there are a set of nodes, one per variable X. And we have a directed and acyclic graph. Each node of this graph shows a conditional distribution. ### Probabilities in Bayes' Nets - in BNs, join distribution is obtained by the product of local conditional distributions. So, to find the probability of a full assignment, multiply all relevant conditional probabilities. + in BNs, joint distribution is obtained by the product of local conditional distributions. So, to find the probability of a full assignment, multiply all relevant conditional probabilities. ![joint](assets/joint.png)
in the following part, we will prove this method is correct.
First note the chain rule applied to each distribution: ![chain](assets/chain.png)
Now with respect to conditional independence, we can claim that: ![conditionaldep](assets/conditionaldep.png)
- Which is true because xi is conditionally independent of other nodes, given its parents.
+ Which is true because xi is conditionally independent of other nodes, given its parents. In other words, by knowing the values of all the parents, the other nodes no longer give us new information, and it does not matter to us whether we know them or not.
So, we can conclude that: ![joint](assets/joint.png)
Note that not every BN can produce every distribution, but the BN topology determines what conditional independence can be produced.
@@ -149,12 +149,12 @@ In this picture there are a set of nodes, one per variable X. And we have a dire For example P(+b,-e,+a,-j,-m)=P(+b)P(-e)P(+a|+b,-e)P(-j|+a)P(-m|+a) ### Causality in Bayes' Nets - If in a Bayes' net the arrows show the real causes, it can be better investigated and the probabilities found. The network also becomes simpler. Because the number of parents is less.
- Sometimes arrows do not show causal relationships. Or even in a network, none of the arrows have a causal relationship. For example, when some important variables are not available in the network. In such cases, the arrows show correlation and not causation.
+ If in a Bayes' net the arrows represent the real causes, it can be better investigated and the probabilities are easier to find. The network also becomes simpler. Because the number of parents is fewer.
+ Sometimes arrows do not represent causal relationships. Or even in some networks, none of the arrows have a causal relationship. For example, when some important variables are not available in the network. In such cases, the arrows show correlation and not causation.
Therefore, we do not need to know the exact causal structure between the variables. ### Space Efficiency - The CPT size of a node if it has m different value and has k parents, each of which also has m different value, is equal to m^(k+1).
+ The CPT size of a node depends on it's domain and parents. if a node has k parents and each of which has m different value, the CPT size is equal to m^(k+1).
Therefore, in a Bayes' net with N Boolean variables, if each node has a maximum of k parents, the size of the whole net is of the order of O(N*2^(k+1)). Whereas if we want to store their joint distribution, it is the size of 2^N.
So, we conclude that if we use this method, it usually requires less memory. It is also easier and faster to use local CPTs. @@ -185,7 +185,8 @@ In this BN, low pressure causes rain and rain causes traffic. in another word ra is X is guaranteed to be independent of Z?
-the answer is NO. To prove this claim, it is enough to give an example that X and Z are dependent. suppose that X is random, and probability of low pressure (P(+x)) and high pressure(P(-x)) are equal to 0.5. P(+x)=P(-x)=0.5
+the answer is NO.
+To prove this claim, it is enough to give an example that X and Z are dependent. suppose that X is random, and probability of low pressure (P(+x)) and high pressure(P(-x)) are equal to 0.5. P(+x)=P(-x)=0.5
rainfall is completely dependent on low pressure and if low pressure occurs, rainfall is definite. and if high pressure occurs, rainfall is impossible.
P(+y|-x)=0 P(+y|+x)=1
@@ -206,7 +207,7 @@ the answer is YES. to prove this claim, we use algebraic operations. we should s we conclude that with condition on Y, X and Z are independent.
-So, in triple casual case, if middle node is observed, end nodes are independent. because the effect of X to Z is not direct, if value of Y be known, change in X’ value causes no effect on Z’ value. in another word, if we have value of Y, value of X does not give any additional information about Z.
+So, in triple casual case, if middle node is observed, end nodes are independent. because the effect of X to Z is not direct, if the value of Y is known, changes in X value causes no effect on Z’ value. in other word if we know value of Y, value of X doesn't give any additional information about Z.
So, evidence along the chain “blocks” the influence. @@ -233,7 +234,7 @@ P(+z|+y)=1 P(+z|-y)=0
Also, forums are related to project deadline. if deadline is near, forums get busy.
P(+x|+y)=1 P(+x|-y)=0
-now we should compare P(Z) and P(Z|X) to find dependency. for example, lab is full and forums are busy. (X=+x and Z=+z) we know P(+z|+x)=1 but P(+z)=0.5 +now we should compare P(Z) and P(Z|X) to find dependency. for example, lab is full and forums are busy. (X=+x and Z=+z) we know P(+z|+x)=1 but P(+z)=0.5. because value of Z and X are exactly equal to the value of Y.
As a result, the two child variables in common cause triples are not independent in general case.
is X is guaranteed to be independent of Z given Y?
@@ -244,7 +245,7 @@ the answer is Yes. to prove this claim, we use algebraic operations. we should s we conclude that with condition on Y, X and Z are independent.
-So, in triple common cause, if parent node is observed, child nodes are independent. because the effect of X to Z is not direct, if value of Y be known, change in X’ value causes no effect on Z’ value. in another word, if we have value of Y, value of X does not give any additional information about Z.
+So, in triple common cause, if parent node is observed, child nodes are independent. because the effect of X to Z is not direct, if value of Y is known, changes in X value causes no effect on Z’ value. in other word if we know value of Y, value of X doesn't give any additional information about Z.
So, observing the cause blocks influence between effects. #### V-Structure @@ -270,7 +271,7 @@ So we show P(X,Y) = P(X)P(Y) and conclude X and Y are independent.
is X is guaranteed to be independent of Y given z?
-the answer is NO. with an example we describe the situation. consider X and Y have random values and probability of occurrence each one is 0.5. (P(+x)=P(+y)=0.5) and traffic will occur when ballgame or raining occurs. +the answer is NO. with an example we describe the situation. consider X and Y have random values and probability of occurrence of each one is 0.5. (P(+x)=P(+y)=0.5) and traffic will occur when ballgame or raining occurs. - P(+z|+x,+y)=P(+z|+x,-y)=P(+z|-x,+y)=1 P(+z|-x,-y)=0
From b467ec52e019089ace561b7182ccf50035035677 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Thu, 30 Dec 2021 02:49:45 +0330 Subject: [PATCH 35/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 12 ++++++++---- 1 file changed, 8 insertions(+), 4 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 57b9bbb6..96c33714 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -277,7 +277,8 @@ the answer is NO. with an example we describe the situation. consider X and Y ha From the problem description, We can conclude that P(+x|+z,-y)=1 and P(+x|-y)=0.5. So, P(X|Z,Y) is not equal to P(X|Y).
-So, with the observation of Z, X and Y are not independent. this means Observing an effect, activating influence between possible causes. +So, with the observation of Z, X and Y are not independent. this means Observing an effect, activating influence between possible causes.
+In the same way, it can be shown that if one of Z descendants is also observed, the independence of X and Y is no longer guaranteed. To prove this from the previous issue, it is enough to add the condition that the occurrence or non-occurrence of each of Z descendants is always the same as its parent. ### Reachability @@ -336,14 +337,14 @@ Examples: ### Structure Implications - As you can see, with the help of this algorithm, we can examine the conditional independence of two random variables. So, if we test the algorithm on all modes, we get a list of conditional independence. but sometimes this list isn’t complete. because when we check conditional dependencies, some cases aren’t certain, and using this method alone, their independence cannot be recognized.
+ As you can see, With the help of this algorithm, sometimes it is possible to check the conditional independence of two random variables more easily. So, if we test the algorithm on all models, we get a list of conditional independence. but sometimes this list isn’t complete. because when we check conditional dependencies, some cases aren’t certain, and using this method alone, their independence cannot be recognized.
but in some cases, “d-separation” algorithm can find all dependencies. For example, in the figure below; ![stIm](assets/sim.jpg) ### Topology Limits Distributions -In a given graph topology, only certain joint distributions can be encoded. The graph structure guarantees certain (conditional) independence (There might be more independence). Adding arcs, increase the dependence of variables.
+In a given graph topology, only certain joint distributions can be encoded. The graph structure guarantees certain (conditional) independence (There might be more independence). As the number of arcs increases, the probability that there is at least one active path between the two variables increases. Thus the independence of the variables decreases.
For example, in the figure below, different kind of dependence for triples is mentioned. ![TLD](assets/tld.jpg) @@ -369,4 +370,7 @@ Up until now, we saw how to build a bayes' net and how to find the independencie ## References
- +
+
+
+ From c87dd1834202ac6d9e6520b01d4d7f883f77a5b3 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Thu, 30 Dec 2021 02:52:55 +0330 Subject: [PATCH 36/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 96c33714..74832c9f 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -185,8 +185,7 @@ In this BN, low pressure causes rain and rain causes traffic. in another word ra is X is guaranteed to be independent of Z?
-the answer is NO.
-To prove this claim, it is enough to give an example that X and Z are dependent. suppose that X is random, and probability of low pressure (P(+x)) and high pressure(P(-x)) are equal to 0.5. P(+x)=P(-x)=0.5
+the answer is NO. To prove this claim, it is enough to give an example that X and Z are dependent. suppose that X is random, and probability of low pressure (P(+x)) and high pressure(P(-x)) are equal to 0.5. P(+x)=P(-x)=0.5
rainfall is completely dependent on low pressure and if low pressure occurs, rainfall is definite. and if high pressure occurs, rainfall is impossible.
P(+y|-x)=0 P(+y|+x)=1
From 0c70b06df888f63f4f937773d9205b3f07c27c2d Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Fri, 31 Dec 2021 20:45:27 +0330 Subject: [PATCH 37/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 74832c9f..a93dd8f7 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -1,5 +1,7 @@ # **Bayesian Networks Representation** +$\frac{n!}{k!(n-k)!}$ + ## **Table of contents:** - [Intoduction](#Intoduction) - [Probabilistic Models](#Probabilistic-Models) From 80a38aa1d1e539b9597fccbc227269bacfe0c9a2 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Fri, 31 Dec 2021 20:47:45 +0330 Subject: [PATCH 38/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index a93dd8f7..bbfd4c7f 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -1,6 +1,6 @@ # **Bayesian Networks Representation** -$\frac{n!}{k!(n-k)!}$ +![\Large x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}] ## **Table of contents:** - [Intoduction](#Intoduction) From 15296a94b80b4fc05d4ed12cef7876145aae2055 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Fri, 31 Dec 2021 20:48:03 +0330 Subject: [PATCH 39/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index bbfd4c7f..0e3d7ba1 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -1,6 +1,6 @@ # **Bayesian Networks Representation** -![\Large x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}] +![\Large x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}](https://latex.codecogs.com/svg.latex?\Large&space;x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}) ## **Table of contents:** - [Intoduction](#Intoduction) From 5c46acc7408644c30ba7214fe2fe3b25c5c12c02 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Fri, 31 Dec 2021 20:56:51 +0330 Subject: [PATCH 40/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 1 + 1 file changed, 1 insertion(+) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 0e3d7ba1..9cc4c23f 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -1,6 +1,7 @@ # **Bayesian Networks Representation** ![\Large x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}](https://latex.codecogs.com/svg.latex?\Large&space;x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}) +$$ E = mc^2 $$ ## **Table of contents:** - [Intoduction](#Intoduction) From 3ed7aa08685931c983e094465962d59500abce10 Mon Sep 17 00:00:00 2001 From: GildedJson Date: Fri, 31 Dec 2021 20:44:32 -0800 Subject: [PATCH 41/59] Added size estimate --- notebooks/bayesian_networks_representation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 74832c9f..c3b28813 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -99,7 +99,7 @@ These independencies can be derived from bayes' nets. This will come in handy wh ### Problems with joint distribution tables There are two problems with using full joint distribution tables as our probabilistic models: -1. Unless there are only a few variables, the joint is WAY too big to represent explicitly. +1. Unless there are only a few variables, the joint is WAY too big to represent explicitly (With 2 to the power of n rows from n variables). 2. Hard to learn (estimate) anything empirically about more than a few variables at a time. Bayes nets is a technique for describing complex joint distributions or models using simple and local distributions. From b2250d2e7ac9e6c45aa67e0f08449d256f5f1adc Mon Sep 17 00:00:00 2001 From: GildedJson Date: Fri, 31 Dec 2021 20:53:11 -0800 Subject: [PATCH 42/59] Added explanation for joint prob tables --- notebooks/bayesian_networks_representation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index c3b28813..00c2f8f7 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -100,7 +100,7 @@ These independencies can be derived from bayes' nets. This will come in handy wh There are two problems with using full joint distribution tables as our probabilistic models: 1. Unless there are only a few variables, the joint is WAY too big to represent explicitly (With 2 to the power of n rows from n variables). -2. Hard to learn (estimate) anything empirically about more than a few variables at a time. +2. Hard to learn (estimate) anything empirically about more than a few variables at a time. For example, to find the probability of two variables being true at the same time, we would have to calculate a sum over n/4 rows which can be an unreasonably big calculation. Bayes nets is a technique for describing complex joint distributions or models using simple and local distributions. From af181c2a9df430b0830c0047593c06907a3a7d7b Mon Sep 17 00:00:00 2001 From: GildedJson Date: Fri, 31 Dec 2021 21:25:16 -0800 Subject: [PATCH 43/59] Fixed some more grammar --- notebooks/bayesian_networks_representation/index.md | 10 ++++++---- 1 file changed, 6 insertions(+), 4 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 00c2f8f7..2413f809 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -106,8 +106,10 @@ Bayes nets is a technique for describing complex joint distributions or models u ### Graphical Notation -- There are some nodes related to variables (with domains). This nodes can be assigned (observed) or unassigned (unobserved) -- There are some arcs equales to interactions. This nodes are similar to CSP constraints and indicate direct influence between variables. And there are formally, it means the ares encode conditional independence. +Bayesian net's graphical notation consists of a graph, and one conditional probability table for each dependency. + +- Each graph node represents a variable . Each variable can be either assigned (observed) or unassigned (unobserved). Note that being observed or not has nothing to do with the variable's domain being binary or not. +- Each graph arc represents an interaction. These arcs are pretty similar to what we saw in previous lecture notes about CSP constraints which indicate direct influence between variables. And there are formally, it means the arcs encode conditional independence. ![image](https://user-images.githubusercontent.com/44923177/146651162-96c91d1f-9f38-4825-9d6f-983f36aca273.png) @@ -118,9 +120,9 @@ Bayes nets is a technique for describing complex joint distributions or models u In this picture there are a set of nodes, one per variable X. And we have a directed and acyclic graph. Each node of this graph shows a conditional distribution. ### Probabilities in Bayes' Nets - in BNs, joint distribution is obtained by the product of local conditional distributions. So, to find the probability of a full assignment, multiply all relevant conditional probabilities. + In BNs, joint distribution is obtained by the product of local conditional distributions. So, to find the probability of a full assignment, multiply all relevant conditional probabilities. ![joint](assets/joint.png)
- in the following part, we will prove this method is correct.
+ In the following part, we will prove this method is correct.
First note the chain rule applied to each distribution: ![chain](assets/chain.png)
Now with respect to conditional independence, we can claim that: From 748d1ed3ae2a39b7b7c680f968c715ed0a29a8b1 Mon Sep 17 00:00:00 2001 From: GildedJson Date: Fri, 31 Dec 2021 21:39:23 -0800 Subject: [PATCH 44/59] Fixed one to once --- notebooks/bayesian_networks_representation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index b08e1db7..f74c2bd6 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -36,7 +36,7 @@ Imagine this scenario. You want to know if the food in your fridge is going to s ### Probabilistic Models -So, what is a probabilistic model? A model is a simplification of the real world; it describes how portions of the world work. Since models are simplifications, they do not contain all the details and may not account for every variable or interaction between variables. As George E. P. Box, a British statistician one said: "All models are wrong; but some are useful." +So, what is a probabilistic model? A model is a simplification of the real world; it describes how portions of the world work. Since models are simplifications, they do not contain all the details and may not account for every variable or interaction between variables. As George E. P. Box, a British statistician once said: "All models are wrong; but some are useful." We use probabilistic models to reason about unknown variables, given evidences. This can be done in three ways: From 6c9b687a450132e2e0285994f5f989051f9f073d Mon Sep 17 00:00:00 2001 From: MohammadHosein Gheisarieh <44923177+mhgheisarieh@users.noreply.github.com> Date: Fri, 31 Dec 2021 23:39:54 +0330 Subject: [PATCH 45/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 3 --- 1 file changed, 3 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index f74c2bd6..8193a872 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -1,8 +1,5 @@ # **Bayesian Networks Representation** -![\Large x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}](https://latex.codecogs.com/svg.latex?\Large&space;x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}) -$$ E = mc^2 $$ - ## **Table of contents:** - [Intoduction](#Intoduction) - [Probabilistic Models](#Probabilistic-Models) From c28075e4deb25c2f048ec53c680d0a42bc8c2dd7 Mon Sep 17 00:00:00 2001 From: GildedJson <60608967+GildedJson@users.noreply.github.com> Date: Fri, 31 Dec 2021 12:25:48 -0800 Subject: [PATCH 46/59] Fixed my formulas --- .../bayesian_networks_representation/index.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 8193a872..81b12551 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -48,15 +48,15 @@ We use probabilistic models to reason about unknown variables, given evidences. Two variables are *independent* if: -![formula1](assets/T_Formula1.PNG) +$$\forall x, y : P(x, y)=P(x)P(y)$$ If two variables are independent, their joint distribution is the product of their distributions. This can also be shown in another way: -![formula2](assets/T_Formula2.PNG) +$$\forall x, y : P(x\mid y)=P(x)$$ We show independency as: -![formula3](assets/T_Formula3.PNG) +$$X\perp Y$$ Note that real life joint distributions are at best close to independent. Independence is a *modelling assumption*. We'll get back to assumptions later. @@ -66,21 +66,21 @@ Unconditional (absolute) independence is very rare to come by, because the varia X is conditionally independent of Y given Z -![formula4](assets/T_Formula4.PNG) +$$X\perp Y\mid Z$$ if and only if -![formula5](assets/T_Formula5.PNG) +$$\forall x,y,z:P(x,y\mid z)=P(x\mid z)P(y\mid z)$$ or equivalently, if and only if -![formula6](assets/T_Formula6.PNG) +$$\forall x,y,z:P(x\mid z,y)=P(x\mid z)$$ ### Chain Rule We all know the basic form of chain rule: -![formula7](assets/T2_Formula7.PNG) +$$P(X_1,X_2,...,X_n)=P(X_1)P(X_2\mid X_1)P(X_3\mid X_1,X_2)...$$ Imagine we want P(X, Y, Z). Using the chain rule, we can write: From 3065acaf807009d07a5a0c1c446d4106cdcf144b Mon Sep 17 00:00:00 2001 From: GildedJson <60608967+GildedJson@users.noreply.github.com> Date: Fri, 31 Dec 2021 12:29:07 -0800 Subject: [PATCH 47/59] Quick fixed the "Formally" --- notebooks/bayesian_networks_representation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 81b12551..28034277 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -109,7 +109,7 @@ Bayes nets is a technique for describing complex joint distributions or models u Bayesian net's graphical notation consists of a graph, and one conditional probability table for each dependency. - Each graph node represents a variable . Each variable can be either assigned (observed) or unassigned (unobserved). Note that being observed or not has nothing to do with the variable's domain being binary or not. -- Each graph arc represents an interaction. These arcs are pretty similar to what we saw in previous lecture notes about CSP constraints which indicate direct influence between variables. And there are formally, it means the arcs encode conditional independence. +- Each graph arc represents an interaction. These arcs are pretty similar to what we saw in previous lecture notes about CSP constraints which indicate direct influence between variables. Formally, the arcs encode conditional dependence. ![image](https://user-images.githubusercontent.com/44923177/146651162-96c91d1f-9f38-4825-9d6f-983f36aca273.png) From b5478668c2a001e114fe4cb42c782e58d7bfb7ab Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 1 Jan 2022 00:12:21 +0330 Subject: [PATCH 48/59] Update index.md --- .../bayesian_networks_representation/index.md | 126 +++++++++--------- 1 file changed, 63 insertions(+), 63 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 28034277..761d8cea 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -120,16 +120,16 @@ Bayesian net's graphical notation consists of a graph, and one conditional proba In this picture there are a set of nodes, one per variable X. And we have a directed and acyclic graph. Each node of this graph shows a conditional distribution. ### Probabilities in Bayes' Nets - In BNs, joint distribution is obtained by the product of local conditional distributions. So, to find the probability of a full assignment, multiply all relevant conditional probabilities. -![joint](assets/joint.png)
+ In BNs, joint distribution is obtained by the product of local conditional distributions. So, to find the probability of a full assignment, multiply all relevant conditional probabilities.
+ $$P(x_1, x_2,... , x_n)=\prod_{i=1}^{n}P(x_i|Parents(x_i))$$
In the following part, we will prove this method is correct.
- First note the chain rule applied to each distribution: -![chain](assets/chain.png)
- Now with respect to conditional independence, we can claim that: -![conditionaldep](assets/conditionaldep.png)
+ First note the chain rule applied to each distribution:
+ $$P(x_1, x_2,... , x_n)=\prod_{i=1}^{n}P(x_i|x_1,x_2,...,x_{i-1})$$
+ Now with respect to conditional independence, we can claim that:
+ $$P(x_i|x_1,x_2,...,x_{i-1})= P(x_i|Parents(X_i))$$
Which is true because xi is conditionally independent of other nodes, given its parents. In other words, by knowing the values of all the parents, the other nodes no longer give us new information, and it does not matter to us whether we know them or not.
- So, we can conclude that: -![joint](assets/joint.png)
+ So, we can conclude that:
+ $$P(x_1, x_2,... , x_n)=\prod_{i=1}^{n}P(x_i|Parents(x_i))$$
Note that not every BN can produce every distribution, but the BN topology determines what conditional independence can be produced.
See the following examples of BN distribution.
@@ -137,18 +137,18 @@ In this picture there are a set of nodes, one per variable X. And we have a dire a. coin flips
In this case, a coin is tossed n times, the probability of a head or tail being equal in each toss. If these actions are independent of each other, then the graph is as shown in the image. ![coin](assets/coin.png)
- For example P(H, T, T, H) = P(H)P(T)P(T)P(H)
+ For example $P(H, T, T, H) = P(H)P(T)P(T)P(H)$
Note that only distributions whose variables are absolutely independent can be represented by a Bayes’ net with no arcs.
b. traffic
In this example, R stands for rain and T stands for traffic. It is also assumed that rain causes traffic. ![traffic](assets/traffic.png)
- For example P(+r, -t) = P(+r)P(-t|+r)
+ For example $P(+r, -t) = P(+r)P(-t|+r)$
c. alarm network
In this case, it is assumed that the house alarm goes off by an earthquake or burglary. John and Mary may also call us if the alarm is ringing. ![alarm](assets/alarm.png)
-For example P(+b,-e,+a,-j,-m)=P(+b)P(-e)P(+a|+b,-e)P(-j|+a)P(-m|+a) +For example $P(+b,-e,+a,-j,-m)=P(+b)P(-e)P(+a|+b,-e)P(-j|+a)P(-m|+a)$ ### Causality in Bayes' Nets If in a Bayes' net the arrows represent the real causes, it can be better investigated and the probabilities are easier to find. The network also becomes simpler. Because the number of parents is fewer.
@@ -156,8 +156,8 @@ For example P(+b,-e,+a,-j,-m)=P(+b)P(-e)P(+a|+b,-e)P(-j|+a)P(-m|+a) Therefore, we do not need to know the exact causal structure between the variables. ### Space Efficiency - The CPT size of a node depends on it's domain and parents. if a node has k parents and each of which has m different value, the CPT size is equal to m^(k+1).
- Therefore, in a Bayes' net with N Boolean variables, if each node has a maximum of k parents, the size of the whole net is of the order of O(N*2^(k+1)). Whereas if we want to store their joint distribution, it is the size of 2^N.
+ The CPT size of a node depends on it's domain and parents. if a node has $k$ parents and each of which has $m$ different value, the CPT size is equal to $m^{k+1}$.
+ Therefore, in a Bayes' net with $N$ Boolean variables, if each node has a maximum of $k$ parents, the size of the whole net is of the order of $O(N\times 2^{k+1})$. Whereas if we want to store their joint distribution, it is the size of $2^N$.
So, we conclude that if we use this method, it usually requires less memory. It is also easier and faster to use local CPTs. @@ -181,34 +181,34 @@ In this configuration, middle node transmits the effect of the previous node to Example: ![causal_ex](assets/ch1.png) -- P(X, Y, Z) = P(X) P(Y|X) P(Z|Y)
+- $$P(X, Y, Z) = P(X) P(Y|X) P(Z|Y)$$
In this BN, low pressure causes rain and rain causes traffic. in another word rain transmits the effect of low pressure to traffic.
-is X is guaranteed to be independent of Z?
+is $X$ is guaranteed to be independent of $Z$?
-the answer is NO. To prove this claim, it is enough to give an example that X and Z are dependent. suppose that X is random, and probability of low pressure (P(+x)) and high pressure(P(-x)) are equal to 0.5. P(+x)=P(-x)=0.5
+the answer is NO. To prove this claim, it is enough to give an example that $X$ and $Z$ are dependent. suppose that $X$ is random, and probability of low pressure $(P(+x))$ and high pressure $(P(-x))$ are equal to 0.5. $P(+x)=P(-x)=0.5$
rainfall is completely dependent on low pressure and if low pressure occurs, rainfall is definite. and if high pressure occurs, rainfall is impossible.
-P(+y|-x)=0 P(+y|+x)=1
+$$P(+y|-x)=0 , P(+y|+x)=1$$
Also, traffic is completely dependent on rainfall. if it rains, traffic will occur and otherwise there isn’t any traffic.
-P(+z|+x)=1 P(-z|-x)= 1
+$$P(+z|+x)=1 , P(-z|-x)=1$$
-now we should compare P(Z) and P(Z|X) to find dependency. for example low pressure and traffic occurred. (X=+x and Z=+z) we know P(+z|+x)=1 but P(+z)=0.5
+now we should compare $P(Z)$ and $P(Z|X)$ to find dependency. for example low pressure and traffic occurred. $(X=+x and Z=+z)$ we know $P(+z|+x)=1$ but $P(+z)=0.5$
As a result, the two variables are not independent.
-is X is guaranteed to be independent of Z given Y?
+is $X$ is guaranteed to be independent of $Z$ given $Y$?
-the answer is YES. to prove this claim, we use algebraic operations. we should show equality of P(Z|Y) and P(Z|Y,X). +the answer is YES. to prove this claim, we use algebraic operations. we should show equality of $P(Z|Y)$ and $P(Z|Y,X)$.
-![ch_formula1](assets/ch10.PNG) +$$P(Z|X,Y)=\frac{P(X,Y,Z)}{P(X,Y)}=\frac{P(X)P(Y|X)P(Z|Y)}{P(X)P(Y|X)}=P(Z|Y)$$
-we conclude that with condition on Y, X and Z are independent.
+we conclude that with condition on $Y$, $X$ and $Z$ are independent.
-So, in triple casual case, if middle node is observed, end nodes are independent. because the effect of X to Z is not direct, if the value of Y is known, changes in X value causes no effect on Z’ value. in other word if we know value of Y, value of X doesn't give any additional information about Z.
+So, in triple casual case, if middle node is observed, end nodes are independent. because the effect of $X$ to $Z$ is not direct, if the value of $Y$ is known, changes in $X$ value causes no effect on Z’ value. in other word if we know value of $Y$, value of $X$ doesn't give any additional information about $Z$.
So, evidence along the chain “blocks” the influence. @@ -221,32 +221,32 @@ in this configuration one variable effect on another two variables. but child no Example: ![common_cause_ex](assets/ch2.jpg) -- P(X, Y, Z) = P(Y) P(X|Y) P(Z|Y)
+- $$P(X, Y, Z) = P(Y) P(X|Y) P(Z|Y)$$
in this BN, project due causes lab full and forums busy.
-is X is guaranteed to be independent of Z?
+is $X$ is guaranteed to be independent of $Z$?
-the answer is NO. it is enough to give an example set of CPTs that X and Z are dependent. suppose that Y is random, and the probability of being close to project deadline (P(+y)) and otherwise (P(-y)) are equal to 0.5. P(+y)=P(-y)=0.5
+the answer is NO. it is enough to give an example set of CPTs that $X$ and $Z$ are dependent. suppose that $Y$ is random, and the probability of being close to project deadline $(P(+y))$ and otherwise $(P(-y))$ are equal to 0.5. $P(+y)=P(-y)=0.5$
laboratory fullness is completely dependent on project deadline and if deadline is near, laboratory is full. and if project deadline is not near, laboratory is not full.
-P(+z|+y)=1 P(+z|-y)=0
+$P(+z|+y)=1 , P(+z|-y)=0$
Also, forums are related to project deadline. if deadline is near, forums get busy.
-P(+x|+y)=1 P(+x|-y)=0
+$P(+x|+y)=1 , P(+x|-y)=0$
-now we should compare P(Z) and P(Z|X) to find dependency. for example, lab is full and forums are busy. (X=+x and Z=+z) we know P(+z|+x)=1 but P(+z)=0.5. because value of Z and X are exactly equal to the value of Y.
+now we should compare $P(Z)$ and $P(Z|X)$ to find dependency. for example, lab is full and forums are busy. $(X=+x and Z=+z)$ we know $P(+z|+x)=1$ but $P(+z)=0.5$. because value of $Z$ and $X$ are exactly equal to the value of $Y$.
As a result, the two child variables in common cause triples are not independent in general case.
-is X is guaranteed to be independent of Z given Y?
+is $X$ is guaranteed to be independent of $Z$ given $Y$?
-the answer is Yes. to prove this claim, we use algebraic operations. we should show equality of P(Z|Y) and P(Z|Y,X). +the answer is Yes. to prove this claim, we use algebraic operations. we should show equality of $P(Z|Y)$ and $P(Z|X,Y)$.
-![ch_formula2](assets/ch11.PNG) +$$P(Z|X,Y)=\frac{P(X,Y,Z)}{P(X,Y)}=\frac{P(X)P(Y|X)P(Z|Y)}{P(X)P(Y|X)}=P(Z|Y)$$
-we conclude that with condition on Y, X and Z are independent.
+we conclude that with condition on $Y$, $X$ and $Z$ are independent.
-So, in triple common cause, if parent node is observed, child nodes are independent. because the effect of X to Z is not direct, if value of Y is known, changes in X value causes no effect on Z’ value. in other word if we know value of Y, value of X doesn't give any additional information about Z.
+So, in triple common cause, if parent node is observed, child nodes are independent. because the effect of $X$ to $Z$ is not direct, if value of $Y$ is known, changes in $X$ value causes no effect on $Z$’ value. in other word if we know value of $Y$, value of $X$ doesn't give any additional information about $Z$.
So, observing the cause blocks influence between effects. #### V-Structure @@ -258,28 +258,28 @@ in this configuration, one variable is affected by another two variables.
Example: ![common_effect_ex](assets/ch3.jpg) -- P(X, Y, Z) = P(X) P(Y) P(Z|X, Y)
+- $$P(X, Y, Z) = P(X) P(Y) P(Z|X, Y)$$
in following BN, both ballgame and raining cause traffic.
-is X is guaranteed to be independent of Y?
+is $X$ is guaranteed to be independent of $Y$?
-the answer, unlike previous versions, is Yes. ballgame and rain do not get effect from each other or a common cause. to prove this claim, use algebraic operation again: +the answer, unlike previous versions, is Yes. ballgame and rain do not get effect from each other or a common cause. to prove this claim, use algebraic operation again:
-![ch_formula1](assets/ch12.PNG) +$$P(X,Y)=\sum_{z}^{}P(X,Y,z)=\sum_{z}^{}P(X)P(Y)P(z|X,Y)=P(X)P(Y)\sum_{z}^{}P(z|X,Y)=P(X)P(Y)$$
-So we show P(X,Y) = P(X)P(Y) and conclude X and Y are independent.
+So we show $P(X,Y) = P(X)P(Y)$ and conclude $X$ and $Y$ are independent.
-is X is guaranteed to be independent of Y given z?
+is $X$ is guaranteed to be independent of $Y$ given $Z$?
-the answer is NO. with an example we describe the situation. consider X and Y have random values and probability of occurrence of each one is 0.5. (P(+x)=P(+y)=0.5) and traffic will occur when ballgame or raining occurs. +the answer is NO. with an example we describe the situation. consider $X$ and $Y$ have random values and probability of occurrence of each one is 0.5. $(P(+x)=P(+y)=0.5)$ and traffic will occur when ballgame or raining occurs.
-- P(+z|+x,+y)=P(+z|+x,-y)=P(+z|-x,+y)=1 P(+z|-x,-y)=0
+$P(+z|+x,+y)=P(+z|+x,-y)=P(+z|-x,+y)=1 , P(+z|-x,-y)=0$
-From the problem description, We can conclude that P(+x|+z,-y)=1 and P(+x|-y)=0.5. So, P(X|Z,Y) is not equal to P(X|Y).
+From the problem description, We can conclude that $P(+x|+z,-y)=1$ and $P(+x|-y)=0.5$. So, $P(X|Z,Y)$ is not equal to $P(X|Y)$.
-So, with the observation of Z, X and Y are not independent. this means Observing an effect, activating influence between possible causes.
-In the same way, it can be shown that if one of Z descendants is also observed, the independence of X and Y is no longer guaranteed. To prove this from the previous issue, it is enough to add the condition that the occurrence or non-occurrence of each of Z descendants is always the same as its parent. +So, with the observation of $Z$, $X$ and $Y$ are not independent. this means Observing an effect, activating influence between possible causes.
+In the same way, it can be shown that if one of $Z$ descendants is also observed, the independence of $X$ and $Y$ is no longer guaranteed. To prove this from the previous issue, it is enough to add the condition that the occurrence or non-occurrence of each of $Z$ descendants is always the same as its parent. ### Reachability @@ -289,19 +289,19 @@ in the first step, we should shade evidence nodes, nodes that are observed in th ### Active/Inactive Paths -Question: are X and Y conditionally independent given evidence variables {Z}?
+Question: are $X$ and $Y$ conditionally independent given evidence variables $Z$?
-Yes, if x and y “d-separated” by z. +Yes, if $X$ and $Y$ “d-separated” by $Z$. to describe the "d-separated" concept, we first need to understand the concept of active/inactive path.
a path is active if each triple of it is active. a triple is active if: -- Causal chain A -> B -> C where B is unobserved (either direction) -- Common cause A <- B -> C where B is unobserved -- Common effect (aka v-structure) A -> B <- C where B or one of its descendants is observed +- Causal chain $A \rightarrow B \rightarrow C$ where $B$ is unobserved (either direction) +- Common cause $A \leftarrow B \rightarrow C$ where $B$ is unobserved +- Common effect (aka v-structure) $A \rightarrow B \leftarrow C$ where $B$ or one of its descendants is observed -Now we consider all undirected paths from X to Y after shading evidence variables, if none of these paths aren’t active paths, we can say X and Y are d-separated by Z.
+Now we consider all undirected paths from $X$ to $Y$ after shading evidence variables, if none of these paths aren’t active paths, we can say $X$ and $Y$ are d-separated by $Z$.
in the figure below, examples of active and inactive triples are shown. @@ -311,30 +311,30 @@ note that a single inactive triple makes a path inactive. but two RVs are indepe Examples: - a. we want to check conditionally independence between R and B variables with different evidence. + a. we want to check conditionally independence between $R$ and $B$ variables with different evidence. ![exp1](assets/ch5.jpg) -- **without evidence:** there is only one path between R and B that is a “common effect” triple. it is inactive so R and B are independent. -- **evidence=T:** there is a “common effect” triple that is active. So, the independence of R and B isn’t guaranteed. +- **without evidence:** there is only one path between $R$ and $B$ that is a “common effect” triple. it is inactive so $R$ and $B$ are independent. +- **evidence=T:** there is a “common effect” triple that is active. So, the independence of $R$ and $B$ isn’t guaranteed. - **evidence=T’:** this situation is similar to previous evidence. - b. we want to check conditionally independence between L and B variables with different evidence. + b. we want to check conditionally independence between $L$ and $B$ variables with different evidence. ![exp2](assets/ch6.jpg) -- **without evidence:** there is only one path between L and B that has two triples. L->R->T is active but R->T<-B is inactive. So, this pass is inactive. So, L and B are independent. -- **evidence = T:** similar to the previous segment there is only one path that has two triples. both are active. So, L and B aren't guaranteed to be independent given T. +- **without evidence:** there is only one path between $L$ and $B$ that has two triples. $L \rightarrow R \rightarrow T$ is active but $R \rightarrow T \leftarrow B$ is inactive. So, this pass is inactive. So, $L$ and $B$ are independent. +- **evidence = T:** similar to the previous segment there is only one path that has two triples. both are active. So, $L$ and $B$ aren't guaranteed to be independent given $T$. - **evidence = T':** it is similar to segment. -- **evidence = T, R:** similar to the 2’nd segment the R->T<-B triple is active but L->R->T is inactive. So, L and B are independent given T and R. +- **evidence = T, R:** similar to the 2’nd segment the $R \rightarrow T \leftarrow B$ triple is active but $L \rightarrow R \rightarrow T$ is inactive. So, $L$ and $B$ are independent given $T$ and $R$. - c. we want to check conditionally independence between T and D variables with different evidence. + c. we want to check conditionally independence between $T$ and $D$ variables with different evidence. ![exp3](assets/ch7.jpg) -- **without evidence:** There are two paths between T and D. upper path is an active “common cause” and the lower path is an inactive “common effect”. So, L and T aren't guaranteed to be independent. -- **evidence = R:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an inactive “common effect”. So, L and T are independent given R. -- **evidence = R, S:** There are two paths between T and D. upper path is an inactive “common cause” and the lower path is an active “common effect”. So, L and T aren't guaranteed to be independent. +- **without evidence:** There are two paths between $T$ and $D$. upper path is an active “common cause” and the lower path is an inactive “common effect”. So, $L$ and $T$ aren't guaranteed to be independent. +- **evidence = R:** There are two paths between $T$ and $D$. upper path is an inactive “common cause” and the lower path is an inactive “common effect”. So, $L$ and $T$ are independent given $R$. +- **evidence = R, S:** There are two paths between $T$ and $D$. upper path is an inactive “common cause” and the lower path is an active “common effect”. So, $L$ and $T$ aren't guaranteed to be independent. ### Structure Implications From 85598f922901166d7151003197144902994243cf Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Sat, 1 Jan 2022 00:17:35 +0330 Subject: [PATCH 49/59] Update index.md --- .../bayesian_networks_representation/index.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 761d8cea..560e24c4 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -64,7 +64,7 @@ Note that real life joint distributions are at best close to independent. Indepe Unconditional (absolute) independence is very rare to come by, because the variables in a problem are usually all correlated. Conditional independence is our most basic and robust form of knowledge about uncertain environments. -X is conditionally independent of Y given Z +$X$ is conditionally independent of $Y$ given $Z$, $$X\perp Y\mid Z$$ @@ -82,13 +82,13 @@ We all know the basic form of chain rule: $$P(X_1,X_2,...,X_n)=P(X_1)P(X_2\mid X_1)P(X_3\mid X_1,X_2)...$$ -Imagine we want P(X, Y, Z). Using the chain rule, we can write: +Imagine we want $P(X, Y, Z)$. Using the chain rule, we can write: -- P(X,Y,Z) = P(X)P(Y|X)P(Z|X,Y) +- $$P(X,Y,Z) = P(X)P(Y|X)P(Z|X,Y)$$ -But if we know that Z and Y are independent, we can instead write: +But if we know that $Z$ and $Y$ are independent, we can instead write: -- P(X,Y,Z) = P(X)P(Y|X)P(Z|X) +- $$P(X,Y,Z) = P(X)P(Y|X)P(Z|X)$$ These independencies can be derived from bayes' nets. This will come in handy when we want to infer probabilities from bayes nets. @@ -100,7 +100,7 @@ These independencies can be derived from bayes' nets. This will come in handy wh There are two problems with using full joint distribution tables as our probabilistic models: 1. Unless there are only a few variables, the joint is WAY too big to represent explicitly (With 2 to the power of n rows from n variables). -2. Hard to learn (estimate) anything empirically about more than a few variables at a time. For example, to find the probability of two variables being true at the same time, we would have to calculate a sum over n/4 rows which can be an unreasonably big calculation. +2. Hard to learn (estimate) anything empirically about more than a few variables at a time. For example, to find the probability of two variables being true at the same time, we would have to calculate a sum over $\frac{n}{4}$ rows which can be an unreasonably big calculation. Bayes nets is a technique for describing complex joint distributions or models using simple and local distributions. @@ -117,7 +117,7 @@ Bayesian net's graphical notation consists of a graph, and one conditional proba ![image](https://user-images.githubusercontent.com/44923177/146654189-38c25839-e446-420f-af85-473900274dfb.png) -In this picture there are a set of nodes, one per variable X. And we have a directed and acyclic graph. Each node of this graph shows a conditional distribution. +In this picture there are a set of nodes, one per variable $X$. And we have a directed and acyclic graph. Each node of this graph shows a conditional distribution. ### Probabilities in Bayes' Nets In BNs, joint distribution is obtained by the product of local conditional distributions. So, to find the probability of a full assignment, multiply all relevant conditional probabilities.
From 878d815aacfde0b7e69aa2df320e87d9e490e5d1 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Tue, 11 Jan 2022 19:55:55 +0330 Subject: [PATCH 50/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 560e24c4..c4e79ad5 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -99,8 +99,8 @@ These independencies can be derived from bayes' nets. This will come in handy wh ### Problems with joint distribution tables There are two problems with using full joint distribution tables as our probabilistic models: -1. Unless there are only a few variables, the joint is WAY too big to represent explicitly (With 2 to the power of n rows from n variables). -2. Hard to learn (estimate) anything empirically about more than a few variables at a time. For example, to find the probability of two variables being true at the same time, we would have to calculate a sum over $\frac{n}{4}$ rows which can be an unreasonably big calculation. +1. Unless there are only a few variables, the joint is WAY too big to represent explicitly (With $2^n$ rows from n variables). +2. Hard to learn (estimate) anything empirically about more than a few variables at a time. For example, to find the probability of two variables being true at the same time, we would have to calculate a sum over $\frac{2^n}{4}$ rows which can be an unreasonably big calculation. Bayes nets is a technique for describing complex joint distributions or models using simple and local distributions. From 76e5f49bc91c2389460d73865430c60210696fdc Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Tue, 11 Jan 2022 19:57:27 +0330 Subject: [PATCH 51/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index c4e79ad5..fb9cb8d3 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -99,7 +99,7 @@ These independencies can be derived from bayes' nets. This will come in handy wh ### Problems with joint distribution tables There are two problems with using full joint distribution tables as our probabilistic models: -1. Unless there are only a few variables, the joint is WAY too big to represent explicitly (With $2^n$ rows from n variables). +1. Unless there are only a few variables, the joint is WAY too big to represent explicitly (With $2^n$ rows from $n$ variables). 2. Hard to learn (estimate) anything empirically about more than a few variables at a time. For example, to find the probability of two variables being true at the same time, we would have to calculate a sum over $\frac{2^n}{4}$ rows which can be an unreasonably big calculation. Bayes nets is a technique for describing complex joint distributions or models using simple and local distributions. From 58c1b77c8c5bf53b8f66d4540eea1b371b8ecf6b Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Tue, 11 Jan 2022 20:06:15 +0330 Subject: [PATCH 52/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index fb9cb8d3..b6e16a38 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -6,7 +6,7 @@ - [Independence](#Independence) - [Conditional Independence](#Conditional-Independence) - [Chain Rule](#Chain-Rule) -- [Bayes' Nets](Bayes'-Nets) +- [Bayes' Nets](Bayes-Nets) - [Problems with joint distribution tables](#Problems-with-joint-distribution-tables) - [Graphical Notation](#Graphical-Notation) - [Semantics](#Semantics) From feda3cbc8881a1cf58733eecd8c9ac75046736b4 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Tue, 11 Jan 2022 20:08:16 +0330 Subject: [PATCH 53/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index b6e16a38..bd763ea1 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -6,11 +6,11 @@ - [Independence](#Independence) - [Conditional Independence](#Conditional-Independence) - [Chain Rule](#Chain-Rule) -- [Bayes' Nets](Bayes-Nets) +- [Bayes' Nets](#Bayes'-Nets) - [Problems with joint distribution tables](#Problems-with-joint-distribution-tables) - [Graphical Notation](#Graphical-Notation) - [Semantics](#Semantics) - - [Probabilities in Bayes' Nets](#Probabilities-in-Bayes'-Nets) + - [Probabilities in Bayes' Nets](#Probabilities-in-Bayes' Nets) - [Causality in Bayes' Nets](#Causality-in-Bayes'-Nets) - [Space Efficiency](#Space-Efficiency) - [Independence in Bayes' Nets](#Independence-in-Bayes'-Nets) From 20ef5901686620d343f92a01892601a3e143a13e Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Tue, 11 Jan 2022 20:14:01 +0330 Subject: [PATCH 54/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index bd763ea1..f23615c1 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -208,7 +208,7 @@ $$P(Z|X,Y)=\frac{P(X,Y,Z)}{P(X,Y)}=\frac{P(X)P(Y|X)P(Z|Y)}{P(X)P(Y|X)}=P(Z|Y)$$ we conclude that with condition on $Y$, $X$ and $Z$ are independent.
-So, in triple casual case, if middle node is observed, end nodes are independent. because the effect of $X$ to $Z$ is not direct, if the value of $Y$ is known, changes in $X$ value causes no effect on Z’ value. in other word if we know value of $Y$, value of $X$ doesn't give any additional information about $Z$.
+So, in triple casual case, if middle node is observed, end nodes are independent. because the effect of $X$ to $Z$ is not direct, if the value of $Y$ is known, changes in value of $X$ causes no effect on Z’ value. in other word if we know value of $Y$, value of $X$ doesn't give any additional information about $Z$.
So, evidence along the chain “blocks” the influence. @@ -246,7 +246,7 @@ $$P(Z|X,Y)=\frac{P(X,Y,Z)}{P(X,Y)}=\frac{P(X)P(Y|X)P(Z|Y)}{P(X)P(Y|X)}=P(Z|Y)$$ we conclude that with condition on $Y$, $X$ and $Z$ are independent.
-So, in triple common cause, if parent node is observed, child nodes are independent. because the effect of $X$ to $Z$ is not direct, if value of $Y$ is known, changes in $X$ value causes no effect on $Z$’ value. in other word if we know value of $Y$, value of $X$ doesn't give any additional information about $Z$.
+So, in triple common cause, if parent node is observed, child nodes are independent. because the effect of $X$ to $Z$ is not direct, if value of $Y$ is known, changes in value of $X$ causes no effect on $Z$’ value. in other word if we know value of $Y$, value of $X$ doesn't give any additional information about $Z$.
So, observing the cause blocks influence between effects. #### V-Structure From a5308630e7decb9b84708242eccbbc85b057218e Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Wed, 12 Jan 2022 01:14:36 +0330 Subject: [PATCH 55/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index f23615c1..87414641 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -6,7 +6,7 @@ - [Independence](#Independence) - [Conditional Independence](#Conditional-Independence) - [Chain Rule](#Chain-Rule) -- [Bayes' Nets](#Bayes'-Nets) +- [Bayes' Nets](#Bayes-'-Nets) - [Problems with joint distribution tables](#Problems-with-joint-distribution-tables) - [Graphical Notation](#Graphical-Notation) - [Semantics](#Semantics) From 806383a5133501ccebfa3eef1891f135675f9b9c Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Wed, 12 Jan 2022 01:14:54 +0330 Subject: [PATCH 56/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 87414641..f23615c1 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -6,7 +6,7 @@ - [Independence](#Independence) - [Conditional Independence](#Conditional-Independence) - [Chain Rule](#Chain-Rule) -- [Bayes' Nets](#Bayes-'-Nets) +- [Bayes' Nets](#Bayes'-Nets) - [Problems with joint distribution tables](#Problems-with-joint-distribution-tables) - [Graphical Notation](#Graphical-Notation) - [Semantics](#Semantics) From c013e141bb590a25caf37c5b6a861cd4abc98816 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Wed, 12 Jan 2022 01:17:15 +0330 Subject: [PATCH 57/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index f23615c1..14d9e3fd 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -6,7 +6,7 @@ - [Independence](#Independence) - [Conditional Independence](#Conditional-Independence) - [Chain Rule](#Chain-Rule) -- [Bayes' Nets](#Bayes'-Nets) +- [Bayes' Nets](#Bayes-Nets) - [Problems with joint distribution tables](#Problems-with-joint-distribution-tables) - [Graphical Notation](#Graphical-Notation) - [Semantics](#Semantics) @@ -92,7 +92,7 @@ But if we know that $Z$ and $Y$ are independent, we can instead write: These independencies can be derived from bayes' nets. This will come in handy when we want to infer probabilities from bayes nets. -## Bayes' Nets +## Bayes' Nets ![fridge](assets/Fridge_net.PNG) From c887a2a4b0a2d300d6cad1608aaa0963e77007ea Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Wed, 12 Jan 2022 01:17:56 +0330 Subject: [PATCH 58/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index 14d9e3fd..d770102f 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -92,7 +92,7 @@ But if we know that $Z$ and $Y$ are independent, we can instead write: These independencies can be derived from bayes' nets. This will come in handy when we want to infer probabilities from bayes nets. -## Bayes' Nets +## Bayes' Nets ![fridge](assets/Fridge_net.PNG) From a4d13b8a7b2b6e5a683e39dc27d7e7a07a813fd2 Mon Sep 17 00:00:00 2001 From: Mohammad Cheraghi <58528991+mohammadch80@users.noreply.github.com> Date: Wed, 12 Jan 2022 01:19:54 +0330 Subject: [PATCH 59/59] Update index.md --- notebooks/bayesian_networks_representation/index.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/notebooks/bayesian_networks_representation/index.md b/notebooks/bayesian_networks_representation/index.md index d770102f..13d05cbe 100644 --- a/notebooks/bayesian_networks_representation/index.md +++ b/notebooks/bayesian_networks_representation/index.md @@ -10,8 +10,8 @@ - [Problems with joint distribution tables](#Problems-with-joint-distribution-tables) - [Graphical Notation](#Graphical-Notation) - [Semantics](#Semantics) - - [Probabilities in Bayes' Nets](#Probabilities-in-Bayes' Nets) - - [Causality in Bayes' Nets](#Causality-in-Bayes'-Nets) + - [Probabilities in Bayes' Nets](#Probabilities-in-Bayes-Nets) + - [Causality in Bayes' Nets](#Causality-in-Bayes-Nets) - [Space Efficiency](#Space-Efficiency) - [Independence in Bayes' Nets](#Independence-in-Bayes'-Nets) - [D-separation](#D-separation) @@ -119,7 +119,7 @@ Bayesian net's graphical notation consists of a graph, and one conditional proba In this picture there are a set of nodes, one per variable $X$. And we have a directed and acyclic graph. Each node of this graph shows a conditional distribution. -### Probabilities in Bayes' Nets +### Probabilities in Bayes' Nets In BNs, joint distribution is obtained by the product of local conditional distributions. So, to find the probability of a full assignment, multiply all relevant conditional probabilities.
$$P(x_1, x_2,... , x_n)=\prod_{i=1}^{n}P(x_i|Parents(x_i))$$
In the following part, we will prove this method is correct.
@@ -150,7 +150,7 @@ In this picture there are a set of nodes, one per variable $X$. And we have a di ![alarm](assets/alarm.png)
For example $P(+b,-e,+a,-j,-m)=P(+b)P(-e)P(+a|+b,-e)P(-j|+a)P(-m|+a)$ -### Causality in Bayes' Nets +### Causality in Bayes' Nets If in a Bayes' net the arrows represent the real causes, it can be better investigated and the probabilities are easier to find. The network also becomes simpler. Because the number of parents is fewer.
Sometimes arrows do not represent causal relationships. Or even in some networks, none of the arrows have a causal relationship. For example, when some important variables are not available in the network. In such cases, the arrows show correlation and not causation.