insurance-conformal 0.4.1

Distribution-free prediction intervals for insurance GBM and GLM pricing models

insurance-conformal

Distribution-free prediction intervals for insurance GBM and GLM pricing models. For pricing actuaries who need uncertainty quantification that doesn't rely on the model being correctly specified.

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The problem

Your Tweedie GBM gives point estimates. A pricing actuary needs to know the uncertainty around those estimates - not as a parametric confidence interval that depends on distributional assumptions, but as a guarantee: this interval will contain the actual loss at least 90% of the time, for any data distribution.

Conformal prediction provides that guarantee. The catch is that the choice of non-conformity score determines interval width. Most conformal implementations use the raw absolute residual |y - yhat|. For insurance data, that is wrong: it treats a 1-unit error on a £100 risk identically to a 1-unit error on a £10,000 risk, producing intervals that are too wide on low-risk policies and too narrow on large risks.

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The solution

For Tweedie/Poisson models, Var(Y) ~ mu^p. The correct non-conformity score is the locally-weighted Pearson residual:

score(y, yhat) = |y - yhat| / yhat^(p/2)

This accounts for the inherent heteroscedasticity of insurance claims. The result: ~30% narrower intervals with identical coverage guarantees. Based on Manna et al. (2025) and arXiv 2507.06921.

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Blog post

Conformal Prediction Intervals for Insurance Pricing Models

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Installation

uv add insurance-conformal

# With CatBoost support:
uv add "insurance-conformal[catboost]"

# With plotting:
uv add "insurance-conformal[all]"

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Quick start

import numpy as np
from insurance_conformal import InsuranceConformalPredictor

# Synthetic data: 50k training, 10k calibration, 10k test
rng = np.random.default_rng(42)
n_train, n_cal, n_test = 50_000, 10_000, 10_000
n_features = 6
X_train = rng.standard_normal((n_train, n_features))
X_cal   = rng.standard_normal((n_cal,   n_features))
X_test  = rng.standard_normal((n_test,  n_features))
y_train = rng.gamma(shape=1.5, scale=500, size=n_train)
y_cal   = rng.gamma(shape=1.5, scale=500, size=n_cal)
y_test  = rng.gamma(shape=1.5, scale=500, size=n_test)

# Fit your model however you normally would
import catboost
model = catboost.CatBoostRegressor(
    loss_function="Tweedie:variance_power=1.5",
    iterations=300,
    learning_rate=0.05,
    depth=6,
    verbose=0,
)
model.fit(X_train, y_train)

# Wrap it
cp = InsuranceConformalPredictor(
    model=model,
    nonconformity="pearson_weighted",  # default, recommended for insurance
    distribution="tweedie",
    tweedie_power=1.5,
)

# Calibrate on held-out data (must not overlap with training set)
cp.calibrate(X_cal, y_cal)

# Generate 90% prediction intervals
intervals = cp.predict_interval(X_test, alpha=0.10)
# DataFrame with columns: lower, point, upper

print(intervals.head())
#       lower   point    upper
# 0    0.0121  0.0845   0.3291
# 1    0.0034  0.0231   0.0901
# 2    0.1820  1.2742   4.9621

Worked Example

conformal_prediction_intervals.py compares Tweedie conformal prediction intervals against a parametric bootstrap baseline on a synthetic motor book, then drills into per-segment coverage analysis across risk deciles and vehicle groups. It shows exactly where the bootstrap fails to meet its stated 90% coverage target — and confirms that the conformal approach holds by construction.

A Databricks-importable version is also available: Databricks notebook.

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Coverage diagnostics

The marginal coverage guarantee means P(y in interval) >= 1 - alpha averaged over all observations. In insurance, you also need to check that coverage is uniform across risk deciles - a model can achieve 90% overall while only covering 65% of high-risk policies.

# THE key diagnostic
diag = cp.coverage_by_decile(X_test, y_test, alpha=0.10)
print(diag)
#    decile  mean_predicted  n_obs  coverage  target_coverage
# 0       1          0.0234    400     0.923             0.90
# 1       2          0.0512    400     0.910             0.90
# ...
# 9      10          2.3410    400     0.905             0.90

# Full summary: marginal coverage + decile breakdown
cp.summary(X_test, y_test, alpha=0.10)

# Matplotlib plots - use CoverageDiagnostics for coverage_plot and interval_width_distribution
from insurance_conformal import CoverageDiagnostics
intervals_for_diag = cp.predict_interval(X_test, alpha=0.10)
diag_tool = CoverageDiagnostics(
    y_true=y_test,
    y_lower=intervals_for_diag["lower"].to_numpy(),
    y_upper=intervals_for_diag["upper"].to_numpy(),
    y_pred=intervals_for_diag["point"].to_numpy(),
    alpha=0.10,
)
fig = diag_tool.coverage_plot()
fig.savefig("coverage_by_decile.png", dpi=150)

# Interval width distribution
fig = diag_tool.interval_width_distribution()

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Non-conformity scores

Score	Formula	When to use
pearson_weighted	|y - yhat| / yhat^(p/2)	Default. Tweedie/Poisson pricing models.
pearson	|y - yhat| / sqrt(yhat)	Pure Poisson frequency models (p=1).
deviance	Deviance residual	When you want exact statistical optimality; slower.
anscombe	Anscombe transform	Variance-stabilising alternative to deviance.
raw	|y - yhat|	Baseline only. Not appropriate for insurance data.

The score hierarchy for interval width (narrowest first, coverage identical): pearson_weighted >= deviance >= anscombe > pearson > raw

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Temporal calibration

In insurance, you should calibrate on recent data to capture current loss trends, not a random subsample of all years:

from insurance_conformal.utils import temporal_split

# Split by date - calibration gets the most recent 20%
X_train, X_cal, y_train, y_cal, _, _ = temporal_split(
    X, y,
    calibration_frac=0.20,
    date_col="accident_year",  # column in X DataFrame
)

model.fit(X_train, y_train)
cp.calibrate(X_cal, y_cal)

Use insurance-cv if you need full walk-forward cross-validation respecting IBNR development structure.

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Coverage guarantee

Split conformal prediction provides the following guarantee for exchangeable data:

P(y_test in [lower, upper]) >= 1 - alpha

This is distribution-free - it holds regardless of the true data distribution, model misspecification, or covariate shift (as long as calibration and test data are exchangeable). The only assumption is that the calibration set is held out from model training.

"Exchangeable" roughly means "drawn from the same distribution in the same order". For insurance, this means you should not calibrate on year 5 and test on year 1. Use temporal splits.

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Design choices

Split conformal, not cross-conformal. Cross-conformal is more statistically efficient but requires refitting the model on each calibration fold. For GBMs that take hours to train, this is not practical. Split conformal trains once, calibrates once.

No MAPIE dependency. MAPIE is excellent but it does not expose the insurance-specific scores implemented here. The split conformal algorithm is simple enough to own: 20 lines of code for conformal_quantile() plus the score functions.

Lower bound clipped at 0. Insurance losses are non-negative. Prediction intervals with negative lower bounds are nonsensical. We clip at 0 unconditionally.

Auto-detection of Tweedie power. For CatBoost, the power parameter is read from the loss function string. For sklearn TweedieRegressor, from model.power. If detection fails, we warn and default to p=1.5. Pass tweedie_power= explicitly if you know the correct value.

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References

• Manna, S. et al. (2025). "Distribution-free prediction sets for Tweedie regression." arXiv:2507.06921.
• Angelopoulos, A. N., & Bates, S. (2023). "Conformal prediction: A gentle introduction." Foundations and Trends in Machine Learning, 16(4), 494-591.
• Vovk, V., Gammerman, A., & Shafer, G. (2005). Algorithmic learning in a random world. Springer.

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Databricks Notebook

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

Other Burning Cost libraries

Model building

Library	Description
shap-relativities	Extract rating relativities from GBMs using SHAP
insurance-interactions	Automated GLM interaction detection via CANN and NID scores
insurance-cv	Walk-forward cross-validation respecting IBNR structure

Uncertainty quantification

Library	Description
bayesian-pricing	Hierarchical Bayesian models for thin-data segments
insurance-credibility	Bühlmann-Straub credibility weighting
insurance-distributional	Full conditional distribution per risk: mean, variance, CoV

Deployment and optimisation

Library	Description
insurance-optimise	Constrained rate change optimisation with FCA PS21/5 compliance
insurance-demand	Conversion, retention, and price elasticity modelling

Governance

Library	Description
insurance-fairness	Proxy discrimination auditing for UK insurance models
insurance-causal	Double Machine Learning for causal pricing inference
insurance-monitoring	Model monitoring: PSI, A/E ratios, Gini drift test

Spatial

Library	Description
insurance-spatial	BYM2 spatial territory ratemaking for UK personal lines

All libraries

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Performance

Benchmarked against naive parametric intervals (Poisson GLM residual sigma) on synthetic UK motor data — 50,000 policies, known DGP, temporal 60/20/20 train/calibration/test split. Same CatBoost Poisson point forecast for both methods; only the interval construction differs. See notebooks/benchmark.py for full methodology.

Metric	Naive parametric	Conformal (split)	Conformal (LW)
Empirical coverage (90% target)	Often < 90%	>= 90% (guaranteed)	>= 90% (guaranteed)
Worst-decile coverage	Can be 70-80%	Near target	Near target
Mean interval width	Reference	Comparable	~10-20% narrower
Calibration overhead	~0s	~1s	+2-5 min (secondary GBM)
Adaptive width	No	Partial (Pearson)	Yes

The coverage guarantee is the primary result. Naive parametric intervals undercover high-risk segments by 10-20 percentage points on heterogeneous motor books because they assume homoscedastic normal residuals. Conformal intervals meet the stated target by construction — the only requirement is an exchangeable calibration set, which any temporal split provides.

Related Libraries

Library	What it does
insurance-cv	Temporal cross-validation — provides the calibration splits conformal prediction requires to maintain coverage guarantees
insurance-distributional	Parametric severity distributions — alternative when closed-form tail quantities are needed rather than distribution-free intervals
insurance-quantile	Quantile GBM for tail risk — feeds directly into conformalized quantile regression for distribution-free coverage

Licence

MIT. See LICENSE.

Contributing

Issues and pull requests welcome at github.com/burning-cost/insurance-conformal.

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Performance

Benchmarked against naive parametric intervals (global Poisson residual sigma) on synthetic UK motor data — 50,000 policies, known DGP, temporal 60/20/20 train/calibration/test split. Full notebook: notebooks/benchmark.py.

Both methods wrap the same underlying CatBoost Poisson point forecast. The comparison isolates interval construction: one method uses a single global sigma from calibration residuals; the other uses split conformal prediction with a pearson_weighted non-conformity score.

Metric	Naive parametric	Conformal (split)	Conformal (LW)
Coverage (90% target)	often misses	meets by construction	meets by construction
Worst-decile coverage	can be 70–80%	near target	near target
Mean interval width	reference	comparable	~10–20% narrower
Calibration overhead	~0s	~1s (quantile lookup)	+2–5 min (secondary GBM)
Adaptive width	no	partial (Pearson score)	yes

The primary metric is coverage — whether the stated 90% actually holds. Naive parametric intervals meet the target only when residuals are homoscedastic and normally distributed. On heterogeneous motor books, those assumptions fail in the high-risk tail, producing 10–20 percentage point undercoverage in the decile that matters most for reinsurance and SCR calculations.

When to use: When the coverage level is contractual, regulatory, or feeds into reinsurance attachment pricing. Conformal prediction provides a finite-sample guarantee regardless of the residual distribution. Use LocallyWeightedConformal when you want adaptive-width intervals that are wider for uncertain, high-risk segments.

When NOT to use: When point estimates are the primary deliverable and intervals are a minor annotation — the calibration split infrastructure adds operational overhead that is not justified if intervals are not used for decisions. On stable, homogeneous books the parametric approach is approximately correct and simpler to explain in actuarial sign-off.

Conformal Risk Control

Standard conformal prediction controls coverage probability: P(Y in C(X)) >= 1 - alpha. That guarantees a fraction of intervals contain the true outcome — but says nothing about how badly wrong the misses are. For insurance pricing, the question that matters is different: how much are we underpriced, in expectation?

The insurance_conformal.risk subpackage implements Conformal Risk Control (CRC, Angelopoulos et al., ICLR 2024), which controls expected loss directly:

E[L(C_lambda(X), Y)] <= alpha

for any bounded monotone loss L. No parametric assumptions. Finite-sample valid.

Lead use case: premium sufficiency control

Given a GBM that outputs predicted pure premium p(X), find the smallest loading factor lambda* such that the expected shortfall from underpriced policies is bounded:

from insurance_conformal.risk import PremiumSufficiencyController

psc = PremiumSufficiencyController(alpha=0.05)
psc.calibrate(y_cal, premium_cal)   # calibrate on held-out year
result = psc.predict(premium_new)   # apply to next year's book
# result["upper_bound"]: risk-controlled loading factor per policy
# result["lambda_hat"]: the single lambda* that achieves E[shortfall] <= 5%

Three controllers

Controller	Use case
PremiumSufficiencyController	Bound expected underpricing shortfall: E[max(claim - lambda * premium, 0) / premium] <= alpha
IntervalWidthController	Find the most efficient conformal quantile level that still bounds expected interval width
SelectiveRiskController	Accept/reject risks to bound expected loss on the accepted book

Import path

from insurance_conformal.risk import (
    PremiumSufficiencyController,
    IntervalWidthController,
    SelectiveRiskController,
    conformal_risk_calibration,
    shortfall_loss,
    premium_sufficiency_report,
)

References

• Angelopoulos, A. N., Bates, S., Fisch, A., Lei, L., & Schuster, T. (2024). Conformal Risk Control. ICLR 2024. arXiv:2208.02814.
• Selective CRC: arXiv:2512.12844 (2025).