insurance-experience

Individual policy-level Bayesian posterior experience rating for insurance pricing.

The primary use case is motor fleet pricing, where there is no NCD scheme and experience rating is done with static modification factors that ignore claims recency, frequency patterns, and portfolio-level variance. This library replaces those spreadsheet mod factors with proper Bayesian credibility that decays old claims, estimates variance from the portfolio, and enforces the balance property.

It also applies to any line where individual claims history should adjust the GLM prior: renewal pricing under FCA PS21/5, commercial liability experience modification, or home insurance repeat-claimant pricing.

The output is a multiplicative credibility factor that slots directly into your existing GLM rating engine:

posterior_premium = prior_premium × credibility_factor

Your GLM does the covariate work. This library does the longitudinal experience work. Neither replaces the other.

Who this is for

Pricing actuaries and data scientists working on:

• Motor fleet experience rating — the core use case. Replaces static mod factors with time-decayed Bayesian updates. No NCD exists for fleets; this fills that gap.
• UK personal lines renewal pricing under FCA PS21/5 (individual experience as a defensible pricing signal)
• Commercial liability experience modification
• Home insurance repeat-claimant pricing

The four model tiers

1. StaticCredibilityModel

Bühlmann-Straub applied at individual policy level. Estimates a single kappa parameter (within/between variance ratio) from the portfolio. Simple, interpretable, fast.

Best for: portfolios with 2–4 years of history, limited computational budget, or actuaries who need to explain the formula to a committee.

2. DynamicPoissonGammaModel

State-space Poisson-gamma with seniority weighting. Parameters p and q control how quickly older claims are discounted. Fitted by empirical Bayes MLE on the portfolio log-likelihood (tractable via negative binomial marginals, no MCMC required).

Reference: Ahn, Jeong, Lu, Wüthrich, "Dynamic Bayesian Credibility", arXiv:2308.16058.

Best for: fleets with 3+ years of history where risk quality genuinely changes over time, or situations where a policy's accident record from 5 years ago should count for less than last year's.

3. SurrogateModel

For non-conjugate Bayesian models where the true posterior requires MCMC. Computes exact Bayesian premiums via importance sampling on a representative sub-portfolio (1–10%), then fits a WLS function g(.) that approximates log(CF) from a sufficient statistic. Scales to large portfolios without per-policy MCMC.

Reference: Calcetero Vanegas, Badescu, Lin, Insurance: Mathematics and Economics 118 (2024).

Best for: when you have a complex severity model (e.g., lognormal with covariates) and the posterior is intractable, but you need to score 100k+ policies.

4. DeepAttentionModel

Linear attention over the claims sequence. Replaces fixed Bühlmann credibility weights with learned weights that can depend on covariates (exposure level, period, claims in that period). Distribution-free fitting via Poisson deviance.

Reference: Wüthrich, "Experience Rating in Insurance Pricing", SSRN 4726206 (2024).

Requires: pip install insurance-experience[deep]

Best for: large portfolios (10k+ policies, 5+ years of history) where the relationship between history and risk is non-linear, and where you have resource to train and validate a neural model.

Installation

pip install insurance-experience

# With deep attention model support (torch required):
pip install insurance-experience[deep]

Quick start

from insurance_experience import ClaimsHistory, StaticCredibilityModel, balance_calibrate

# Describe each policy's claims history
histories = [
    ClaimsHistory(
        policy_id="FLT001",
        periods=[1, 2, 3, 4],
        claim_counts=[0, 1, 0, 0],
        exposures=[1.0, 1.0, 1.0, 0.75],  # years on risk
        prior_premium=12_500.0,             # GLM output
    ),
    ClaimsHistory(
        policy_id="FLT002",
        periods=[1, 2, 3, 4],
        claim_counts=[3, 2, 1, 3],
        exposures=[1.0, 1.0, 1.0, 1.0],
        prior_premium=12_500.0,
    ),
    # ... rest of portfolio
]

# Fit on portfolio (estimates kappa from cross-sectional spread)
model = StaticCredibilityModel()
model.fit(histories)

# Predict credibility factors
cf_good = model.predict(histories[0])   # < 1.0 (better than prior)
cf_bad  = model.predict(histories[1])   # > 1.0 (worse than prior)

# Apply to get posterior premiums
posterior_good = histories[0].prior_premium * cf_good
posterior_bad  = histories[1].prior_premium * cf_bad

# Enforce the balance property: sum(posterior) ≈ sum(actual)
cal = balance_calibrate(model.predict, histories)
print(f"Calibration factor: {cal.calibration_factor:.4f}")

# Production scoring
df = model.predict_batch(histories)
# Returns Polars DataFrame: policy_id, prior_premium, credibility_factor, posterior_premium

Dynamic model with seniority weighting

from insurance_experience import DynamicPoissonGammaModel

model = DynamicPoissonGammaModel()
model.fit(histories)

print(f"Fitted p={model.p_:.3f}, q={model.q_:.3f}")
# p close to 1: strong mean-reversion each period
# q close to 1: older claims retain more weight

# Get uncertainty alongside the point estimate
alpha, beta = model.predict_posterior_params(histories[0])
posterior_mean = alpha / beta
posterior_variance = alpha / beta**2

Surrogate model for large portfolios

from insurance_experience import SurrogateModel

# Sub-portfolio fitting: 5% of policies used for IS calibration
model = SurrogateModel(n_is_samples=2000, subsample_frac=0.05, random_state=42)
model.fit(histories)

# Scoring the full portfolio is instant (analytical g(.) evaluation)
df = model.predict_batch(all_histories)

Balance property

The balance property (Wüthrich 2020, Lindholm & Wüthrich 2025) requires that the sum of posterior premiums equals the sum of observed claims at portfolio level. This is the self-financing constraint: experience rating redistributes the total premium, but does not inflate it.

All models support post-fit balance calibration:

from insurance_experience import balance_calibrate, calibrated_predict_fn, balance_report

cal = balance_calibrate(model.predict, histories)
calibrated = calibrated_predict_fn(model.predict, cal)

# Diagnostic report
report = balance_report(model.predict, histories, by_n_periods=True)
print(report)

Data schema

ClaimsHistory(
    policy_id    = "FLT001",      # str, unique
    periods      = [1, 2, 3],     # list[int], year indices (ascending, unique)
    claim_counts = [0, 1, 0],     # list[int], non-negative
    claim_amounts= [0, 1500, 0],  # list[float] | None (for severity models)
    exposures    = [1.0, 1.0, 0.75],  # list[float], years on risk
    prior_premium= 12_500.0,      # float > 0, GLM output for next period
)

Design choices

Multiplicative credibility factor. All models output CF = posterior/prior rather than the posterior itself. This keeps the library composable: you can replace the GLM, swap the experience model, or apply multiple experience adjustments independently.

sklearn-style interface. fit(histories) then predict(history). No hidden state, no global configuration. Fitted parameters are plain attributes (kappa_, p_, q_) — readable in an audit trail.

Polars-native batch output. predict_batch() returns a Polars DataFrame. At portfolio scale (100k+ policies), Polars is materially faster than pandas for the column operations pricing teams actually do (group-bys by segment, quantile distributions of CF, join to rating factors).

Balance calibration separate from fitting. Some libraries bake calibration into training. We don't. The calibration factor is explicit and auditable. You can see exactly how much the model was rescaled, and in which direction.

torch as optional dependency. The deep attention model requires torch. Requiring torch as a mandatory dependency would be unreasonable for teams who only want Bühlmann-Straub. The [deep] extra makes the dependency explicit.

Relationship to other Burning Cost libraries

• insurance-credibility: Bühlmann-Straub and Hachemeister at group level (fleet, territory, scheme). Use that for pooling across groups. Use this for updating individual policy premiums.
• experience-rating: NCD/bonus-malus Markov chains. That library implements the contractual NCD structure. This library implements the actuarially optimal Bayesian replacement for NCD.

References

• Bühlmann, H. & Gisler, A. (2005). A Course in Credibility Theory. Springer.
• Ahn, J., Jeong, H., Lu, Y. & Wüthrich, M.V. (2023). Dynamic Bayesian credibility. arXiv:2308.16058.
• Calcetero Vanegas, S., Badescu, A. & Lin, X.S. (2024). Effective experience rating via surrogate modelling. Insurance: Mathematics and Economics 118, 25–43. arXiv:2211.06568.
• Wüthrich, M.V. (2024). Experience rating in insurance pricing. SSRN 4726206.
• Wüthrich, M.V. (2020). Bias regularization in neural network models. European Actuarial Journal 10, 179–202.
• Lindholm, M. & Wüthrich, M.V. (2025). The balance property in insurance pricing. Scandinavian Actuarial Journal.

Licence

MIT. See LICENSE.