aLogLikeFit

A collection of flexible routines for first-level and second-level modeling in computational psychiatry, especially for fitting spectral DCMs to neurophysiology data.

This toolkit includes:

• Variational Laplace algorithms with heteroscedastic noise and thermodynamic integration
• Low-rank observation noise modeling
• Parametric Empirical Bayes (PEB) routines for group-level (second-level) Bayesian inference

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\ud83d\udce6 Main Components

1. Variational Laplace Fitters

• fitVariationalLaplaceThermo.m
  Classical Variational Laplace with:

  • Smarter heteroscedastic variance updates
  • Thermodynamic integration for log-evidence estimation
  • Optimizes the ELBO (evidence lower bound)

• fitVL_LowRankNoise.m
  Extended version with:

  • Low-rank plus diagonal modeling of observation noise covariance
  • Dynamic adaptation of noise structure based on residuals

• fitHierarchicalVL.m The fitHierarchicalVL.m function enables hierarchical Bayesian model inversion, allowing for group-level analyses by modeling between-subject variability.

2. Parametric Empirical Bayes (PEB)

• peb_ard_with_stats_var.m, peb_ard_with_stats.m, peb_ard_with_stats_LM.m
  • Implements PEB with:
    • Ridge-like Bayesian regularization
    • Automatic Relevance Determination (ARD) for feature selection
    • (Optional) Levenberg-Marquardt (LM) enhanced optimization for stability
  • Returns:
    • Group-level parameter estimates
    • ARD hyperparameters
    • t-statistics and p-values
    • Individual posteriors

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\ud83d\ude80 Quickstart Guide

Step 1: Fit a First-Level (Individual) Model

Fit a spectral DCM-like model, with fields:
DCM.M.pE, DCM.M.pC, DCM.M.f, DCM.M.IS, DCM.xY

M = aFitDCM(DCM);        % Construct fitting object
M.aloglikVLtherm;        % Run Variational Laplace with thermodynamic integration

Optionally refine:

M.update_parameters(M.Ep); 
M.aloglikVLtherm;

Inspect posteriors:

M.Ep   % Posterior means
M.CP   % Posterior covariances
M.F    % Free energy (model evidence)

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Step 2: Group-Level Inference via PEB

Extract the individual posteriors, and apply PEB:

% Example usage:
[beta, lambda_vals, t_stats, p_values, posterior_means, posterior_covs] = peb_ard_with_stats_var(PosteriorMeans, PosteriorCovariances,X,num_iter);

• G: Group mean parameter estimates
• H: ARD hyperparameters
• stats: t-stats, p-values for group parameters

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\ud83d\udcda Documentation

Each function is documented internally.
See:

• fitVariationalLaplaceThermo.m for classical VL
• fitVL_LowRankNoise.m for noise-structured VL
• peb_ard_with_stats_var.m for second-level modeling.

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\ud83e\uddd0 Why Use aLogLikeFit?

• Designed for neurophysiology DCMs but adaptable to general dynamical system models.
• Supports structured noise \u2014 not just homoscedastic Gaussian assumptions.
• Variational + Empirical Bayes in one lightweight toolbox.
• Minimal dependencies (pure MATLAB).
• Well-suited for small samples and hierarchical modeling.

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\ud83d\udd25 Repository Structure

File	Description
fitVariationalLaplace.m	Basic VL optimizer
fitVariationalLaplaceThermo.m	VL + Thermodynamic integration
fitVL_LowRankNoise.m	VL + Low-rank noise covariance
aFitDCM.m	Simple wrapper for DCM fitting
peb_ard_with_stats_var.m	PEB + ARD (variable noise)
peb_ard_with_stats_LM.m	PEB + ARD + Levenberg-Marquardt

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\ud83d\udce2 Notes

• MATLAB required (tested with R2020b+).
• All code \u00a9 Alexander Shaw 2025.