insurance-distributional-glm 0.1.4

GAMLSS (Generalised Additive Models for Location, Scale and Shape) for insurance pricing in Python

insurance-distributional-glm

GAMLSS (Generalised Additive Models for Location, Scale and Shape) for insurance pricing in Python.

Blog post: GAMLSS in Python, Finally

The problem

Standard GLMs model E[Y|X] — the conditional mean. That's fine when you believe every risk with the same mean also has the same variance. But in motor insurance, a young driver and a middle-aged driver with identical expected claims can have dramatically different claim volatility. Your pricing should know the difference.

GAMLSS fixes this by modelling the full conditional distribution p(Y|X), not just its mean. Each distribution parameter — mean, variance, shape — is expressed as a function of covariates. For a Gamma severity model:

log(mu_i)    = x_i^T beta_mu        # mean depends on risk factors
log(sigma_i) = z_i^T beta_sigma      # CV depends on (possibly different) risk factors

R has had this since 2005 (the gamlss package, 100+ distributions). Python has had nothing production-ready. This fills that gap.

Why this matters for insurance pricing

1. Heterogeneous variance: risks with the same expected loss have different volatility. A high-CV risk needs a different loading than a low-CV risk even if their means are equal.

2. Regulatory pressure: PRA and FCA increasingly expect firms to demonstrate they understand uncertainty in their estimates, not just point predictions. Modelling sigma as a function of covariates is the right answer.

3. Tail behaviour: for commercial lines and liability, the shape of the distribution (not just its mean) drives large loss exposure. Getting sigma right matters more than squeezing another point of fit on the mean.

4. Zero-inflated counts: ZIP models let you separate structural zeros (non-claimants, seasonal risks) from Poisson claim frequency without ad-hoc adjustments.

Installation

uv add insurance-distributional-glm

With matplotlib for diagnostic plots:

uv add "insurance-distributional-glm[plots]"

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Quick start

import numpy as np
import polars as pl
from insurance_distributional_glm import DistributionalGLM
from insurance_distributional_glm.families import Gamma

# Claim severity data
df = pl.DataFrame({
    "age_band": [0.0, 1.0, 2.0, 0.0, 1.0] * 200,          # young / mid / mature
    "vehicle_value": [8000.0, 15000.0, 25000.0] * 333 + [8000.0],
})
rng = np.random.default_rng(42)
y = rng.gamma(4.0, 500.0, len(df))

# Model mean with age + vehicle_value, variance with age only
model = DistributionalGLM(
    family=Gamma(),
    formulas={
        "mu":    ["age_band", "vehicle_value"],
        "sigma": ["age_band"],
    },
)
model.fit(df, y)
model.summary()

Output (exact values depend on random seed and data):

DistributionalGLM — Gamma
  n = 1000, loglik = -7412.xxxx
  Converged: True
  GAIC(2): 14840.xxxx

  Parameter: mu  (link: log)
  Term                            Coef
  --------------------------------------------
  (Intercept)                  ~6.09
  age_band                     ~0.02
  vehicle_value                ~0.00001

  Parameter: sigma  (link: log)
  Term                            Coef
  --------------------------------------------
  (Intercept)                 ~-0.66
  age_band                     ~0.01

Families

Family	Parameters	Insurance use
Gamma	mu (mean), sigma (CV)	Claim severity. Most common choice.
LogNormal	mu (log mean), sigma (log sd)	Severity when log(claims) is symmetric.
InverseGaussian	mu, sigma	Heavy-tailed liability severity.
Tweedie(power)	mu, phi	Pure premiums (includes structural zeros).
Poisson	mu	Claim frequency, baseline.
NBI	mu, sigma (overdispersion)	Overdispersed claim counts. Almost always better than Poisson.
ZIP	mu, pi (zero inflation)	Frequency with excess zeros.

Exposure offsets

from insurance_distributional_glm import DistributionalGLM
from insurance_distributional_glm.families import NBI

# Exposure-weighted frequency model
# df, claim_counts, policy_years, and new_df are your portfolio DataFrames/arrays.
model = DistributionalGLM(family=NBI(), formulas={"mu": ["age_band"], "sigma": []})
model.fit(df, claim_counts, exposure=policy_years)

# Predict rate per unit exposure for new business
rates = model.predict(new_df, parameter="mu", exposure=np.ones(len(new_df)))

Model selection

from insurance_distributional_glm import choose_distribution
from insurance_distributional_glm.families import Gamma, LogNormal, InverseGaussian

results = choose_distribution(
    df, y,
    families=[Gamma(), LogNormal(), InverseGaussian()],
    formulas={"mu": ["age_band", "vehicle_value"], "sigma": []},
    penalty=2.0,   # AIC; use np.log(len(y)) for BIC
)

for r in results:
    print(f"{r.family_name}: GAIC={r.gaic:.1f}, converged={r.converged}")
# Gamma: GAIC=14840.7, converged=True
# LogNormal: GAIC=14901.3, converged=True
# InverseGaussian: GAIC=14923.8, converged=True

Relativities

# Actuarial-style output: multiplicative factors per risk factor level
rel = model.relativities(parameter="mu")
print(rel)
# shape: (n_terms, 4)
# columns: param, term, coefficient, relativity, link

For log-linked parameters, relativity = exp(coefficient) — the multiplicative effect on the predicted mean, exactly as actuaries expect from a GLM output.

Diagnostics

from insurance_distributional_glm import quantile_residuals, worm_plot

# Randomised quantile residuals (Dunn & Smyth 1996)
# For a correct model, these should be iid N(0,1)
resids = quantile_residuals(model, df, y, seed=42)

# Worm plot: detrended QQ plot, split by fitted mu quantile
# Requires matplotlib: uv add "insurance-distributional-glm[plots]"
worm_plot(model, df, y, n_groups=4)

Volatility scoring

# CV = sqrt(Var[Y|X]) / E[Y|X] per risk
cv = model.volatility_score(df)

# Flag high-volatility risks (CV > 0.8)
df_scored = df.with_columns(pl.Series("cv", cv))
high_vol = df_scored.filter(pl.col("cv") > 0.8)

The RS algorithm

Fitting uses the Rigby-Stasinopoulos (RS) algorithm: cycle through each distribution parameter, update it via IRLS (weighted least squares) while holding all others fixed. Convergence criterion is change in total log-likelihood < tol.

This is equivalent to a coordinate descent on the joint log-likelihood, where each coordinate step has a closed-form weighted least squares solution. It's not the most efficient algorithm (CG — Conjugate Gradient — is faster for large p), but it's robust and straightforward to implement correctly.

Design choices

Why numpy/scipy only, no torch? Insurance pricing teams typically work in SQL/Python without GPU infrastructure. A numpy implementation is deployable anywhere.

Why not statsmodels? We tried. statsmodels' GLM is not designed to be extended to multiple linear predictors cleanly, and the formula interface adds overhead that actuaries don't use. Better to build clean from the RS paper.

Why polars as the primary DataFrame interface? Speed for large portfolio operations, and the expression API makes feature engineering readable. Pandas is supported via duck typing.

Why fix the power parameter p in Tweedie? Profile likelihood over a grid of p values (say 1.2 to 1.8 in steps of 0.1) and pick the best. Treating p as a free parameter inside the RS loop causes numerical instability. We may add fit_tweedie_power() as a wrapper in a future version.

Databricks Notebook

A ready-to-run Databricks notebook benchmarking this library against standard approaches is available in burning-cost-examples.

References

• Rigby, R.A. and Stasinopoulos, D.M. (2005). Generalised additive models for location, scale and shape. JRSS-C, 54(3), 507-554.
• Dunn, P.K. and Smyth, G.K. (1996). Randomized quantile residuals. JCGS, 5(3), 236-244.
• Smyth, G.K. and Jørgensen, B. (2002). Fitting Tweedie's compound Poisson model to insurance claims data. ASTIN Bulletin, 32(1), 143-157.

When not to use

• Point predictions only: if your pricing process consumes only E[Y|X] and variance modelling adds no downstream value (no capital allocation, no risk loading, no interval outputs), a standard GLM is simpler and equally correct.
• Small portfolios (fewer than ~1,000 claims): the sigma submodel needs enough data to identify covariate effects on dispersion. Below this threshold, constant-dispersion GLM is more stable and less likely to overfit the variance structure.
• Pure claim frequency: if your target is claim counts with no overdispersion concern, a standard Poisson GLM is the right tool. NBI adds a parameter that may not be justified on thin data.
• Regulatory interpretability is the binding constraint: if your model must be explainable to a regulator or pricing committee who cannot engage with GAMLSS concepts, the overhead of justifying a two-submodel structure may outweigh the technical benefit.

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Related Libraries

Library	What it does
insurance-distributional	Parametric severity distributions — use when the full GAMLSS framework is not needed and fixed-sigma fits are adequate
insurance-dispersion	Double GLM for covariate-driven dispersion — simpler alternative using IRLS rather than the full RS algorithm
insurance-gam	Generalised Additive Models — smooth non-linear effects in the mean submodel before adding distributional parameters

License

MIT

Performance

Benchmarked against standard Gamma GLM with constant phi (statsmodels) on 25,000 synthetic UK motor severity claims. DGP: CV genuinely depends on vehicle class and distribution channel (vehicle D + broker has ~3x higher CV than vehicle A + direct). 70/30 train/test split. Post-Phase-98 fix numbers (Tweedie IRLS weights, gammaln, and d2l corrections applied). Full script: benchmarks/benchmark_insurance_distributional_glm.py.

Metric	Gamma GLM (constant phi)	GAMLSS (DistributionalGLM)
Gamma deviance (test)	0.2385	0.2385
95% PI coverage (target 0.95)	0.9387	0.9425
Sigma MAE vs true DGP	0.1018	0.0059
Sigma correlation with true DGP	0.000	0.998
Variance calibration MAE	n/a (constant sigma)	0.100
Fit time (25k obs)	0.41s	0.39s

The headline result is the sigma recovery. The Gamma GLM assigns sigma=0.467 to every policy regardless of risk factors. GAMLSS recovers covariate-driven sigma with a correlation of 0.998 against the true DGP values — the model correctly learns that vehicle D broker policies have ~3x the CV of vehicle A direct policies.

Gamma deviance and PI coverage are close because on a 25k dataset with dense confounder structure, the mean model is well-identified by both approaches. The difference becomes material for capital modelling (where distributional shape matters more than mean fit) and for segment-level prediction intervals — a vehicle D broker policy deserves a wider interval than the pooled phi implies.

Fit time is comparable: the RS algorithm converges in similar wall-clock time to a single Gamma GLM on this dataset size, because the per-iteration IRLS steps are cheap.