insurance-monitoring 1.2.2

Model drift detection, calibration monitoring, and sequential A/B testing for insurance pricing models: PSI, Gini drift, Murphy decomposition, mSPRT champion/challenger.

insurance-monitoring

Production model monitoring for UK insurance pricing — PSI, Gini drift, Murphy decomposition, and anytime-valid A/B testing in one package. Tells you whether to redeploy, recalibrate, or refit.

Blog post: Insurance Model Monitoring: Beyond Generic Drift Detection

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The problem

A pricing model that is 15% cheap on young drivers and 15% expensive on mature drivers reads 1.00 at portfolio level — and triggers no alarm. The loss ratio deteriorates twelve months later.

PRA SoP3/24 requires an annual attestation (IMOR) that model governance and monitoring are in place. Consumer Duty (PS22/9) requires ongoing evidence that pricing outcomes are fair across customer groups. Standard monitoring approaches fail both obligations in three specific ways:

• Portfolio-level A/E hides segmental drift. Per-feature distribution shifts and model discrimination drift are invisible to a single headline ratio.
• Repeated monthly testing inflates false positives. Running Hosmer-Lemeshow or A/E tests each month means a perfectly calibrated model will trigger a false alarm roughly 40% of the time within a year at α=0.05.
• Champion/challenger comparisons are run to a pre-specified date. Checking interim results is statistically invalid under classical hypothesis testing — but pricing teams do it anyway.

insurance-monitoring addresses all three. It monitors per-feature distribution shifts (PSI/CSI), discrimination and calibration separately (Murphy decomposition), and runs anytime-valid tests you can check at any point.

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Installation

pip install insurance-monitoring
# or
uv add insurance-monitoring

Dependencies: polars, numpy, scipy, matplotlib

MLflow integration (optional):

pip install insurance-monitoring[mlflow]

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Quick start

import numpy as np
from insurance_monitoring import MonitoringReport

rng = np.random.default_rng(42)
# Training period — well-calibrated model
actual_ref = rng.poisson(0.08, 10_000).astype(float)
predicted_ref = np.full(10_000, 0.08)
# Current period — book has aged, model is stale
actual_cur = rng.poisson(0.11, 5_000).astype(float)
predicted_cur = np.full(5_000, 0.08)
exposure_cur = rng.uniform(0.5, 1.0, 5_000)

report = MonitoringReport(
    reference_actual=actual_ref, reference_predicted=predicted_ref,
    current_actual=actual_cur, current_predicted=predicted_cur,
    exposure=exposure_cur, murphy_distribution="poisson",
)
print(report.recommendation)   # => 'RECALIBRATE' or 'REFIT'
print(report.to_polars())      # flat DataFrame: metric / value / band

See examples/ for fully worked scenarios: model_monitor_quickstart.py (the v1.0.0 three-way decision), UK motor frequency monitoring, home insurance with PITMonitor, and a champion/challenger A/B test.

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Part of the Burning Cost toolkit

Takes the outputs of any fitted pricing model and a stream of actual experience. Feeds drift signals and calibration verdicts into insurance-governance for model risk committee packs. Pairs with insurance-fairness to monitor per-protected-group A/E ratios under Consumer Duty. → See the full stack

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What manual monitoring looks like vs this

Task	Manual approach	insurance-monitoring
Population Stability Index	Excel macro per factor, re-coded each quarter, unweighted	psi() / csi() — exposure-weighted, Polars-native, traffic-light band
Feature drift heatmap	Engineer writes one-off script; no standard thresholds	csi() — one call, all rating factors, documented thresholds
A/E ratio with CI	Custom formula in SQL, no confidence interval	ae_ratio_ci() — Wilson CI, exposure-weighted, RAG status
Discrimination drift	Gini computed ad hoc; no test for statistical significance	gini_drift_test() / GiniDriftBootstrapTest — bootstrap CI, governance plot
RECALIBRATE vs REFIT decision	Actuary judgment call, not documented	ModelMonitor — Gini + GMCB + LMCB bootstrap tests (arXiv 2510.04556), structured three-way decision
Repeated monthly testing inflation	H-L / A/E tested afresh each month; inflated false-positive rate	PITMonitor — anytime-valid, P(ever false alarm | calibrated) ≤ α, forever
Champion/challenger A/B test	Wait for pre-specified sample size; cannot stop early	SequentialTest — mSPRT, valid at every interim check, supports frequency/severity/loss ratio
Drift attribution (which feature explains the performance change?)	PSI-by-eye; no interaction-aware method	DriftAttributor / InterpretableDriftDetector — TRIPODD, FDR control, exposure weighting
Monitoring report for model risk committee	Manual Word document	MonitoringReport.to_polars() — flat DataFrame, writes directly to Delta table

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Features

ModelMonitor — structured three-way decision (v1.0.0)

ModelMonitor is the recommended entry point for teams running quarterly or monthly model reviews. It implements the full two-step procedure from Brauer, Menzel & Wüthrich (2025), arXiv:2510.04556.

import numpy as np
from insurance_monitoring import ModelMonitor

rng = np.random.default_rng(42)
n = 10_000
exposure = rng.uniform(0.5, 1.5, n)
y_hat = rng.gamma(2, 0.05, n)
y_ref = rng.poisson(exposure * y_hat) / exposure   # reference period

monitor = ModelMonitor(distribution="poisson", n_bootstrap=500, random_state=0)
monitor.fit(y_ref, y_hat, exposure)

# Quarterly monitoring run
y_new = rng.poisson(exposure * y_hat * 1.10) / exposure  # 10% claims trend
result = monitor.test(y_new, y_hat, exposure)

print(result.decision)          # 'REDEPLOY' | 'RECALIBRATE' | 'REFIT'
print(result.balance_factor)    # multiply predictions by this for RECALIBRATE
print(result.summary())         # governance-ready paragraph for MRC packs
print(result.to_dict())         # JSON-serialisable — log to MLflow or Delta

Decision rules:

Test fires	Decision	Action
Nothing	REDEPLOY	No action. Re-run next quarter.
GMCB only	RECALIBRATE	Multiply predictions by balance_factor. Hours of work.
Gini or LMCB	REFIT	Rebuild the model. Weeks of work.

The default alpha=0.32 (one-sigma rule) is calibrated for ongoing production monitoring. See examples/model_monitor_quickstart.py for all three scenarios with realistic synthetic data.

MonitoringReport — one call for a complete monitoring run

from insurance_monitoring import MonitoringReport

report = MonitoringReport(
    reference_actual=train_claims,
    reference_predicted=train_predicted,
    current_actual=current_claims,
    current_predicted=current_predicted,
    exposure=current_exposure,
    feature_df_reference=train_features,    # pl.DataFrame
    feature_df_current=current_features,    # pl.DataFrame
    features=["driver_age", "vehicle_age", "ncd_years"],
    murphy_distribution="poisson",          # Murphy decomposition — always available
    gini_bootstrap=True,                    # percentile CI on Gini (v0.6.0)
)
print(report.recommendation)   # 'NO_ACTION' | 'RECALIBRATE' | 'REFIT' | 'INVESTIGATE'
print(report.to_dict())        # nested dict — JSON serialisable, log to MLflow
print(report.to_polars())      # flat DataFrame — write to Delta table

Recommendation logic follows the three-stage decision tree from arXiv 2510.04556:

• Gini OK + A/E OK → NO_ACTION
• A/E bad only → RECALIBRATE (update intercept)
• Gini bad → REFIT (rebuild on recent data)
• Murphy decomposition present: overrides the heuristic when miscalibration vs discrimination are unambiguous

PSI / CSI — feature distribution monitoring

from insurance_monitoring.drift import psi, csi

# PSI on a single feature — exposure-weighted
drift_val = psi(
    reference=driver_age_train,
    current=driver_age_now,
    n_bins=10,
    exposure_weights=exposure_now,    # car-years, not policy count
    reference_exposure=exposure_train,
)
print(f"PSI: {drift_val:.3f}")   # < 0.10 green | 0.10–0.25 amber | > 0.25 red

# CSI heatmap across all rating factors — returns pl.DataFrame with band column
csi_df = csi(
    reference_df=train_df,
    current_df=current_df,
    features=["driver_age", "vehicle_age", "ncd_years", "vehicle_group"],
)
print(csi_df)  # feature | csi | band

Use PSI/CSI for operational dashboards. Use ks_test for formal hypothesis testing at quarter-end (note: over-sensitive at n > 500k). Use wasserstein_distance when communicating to non-technical stakeholders — it reports drift in original feature units.

Calibration — A/E, Murphy decomposition, anytime-valid PITMonitor

from insurance_monitoring import ae_ratio_ci, murphy_decomposition, PITMonitor
from scipy.stats import poisson

# A/E ratio with Wilson CI
result = ae_ratio_ci(actual, predicted, exposure=exposure)
print(f"A/E = {result['ae']:.3f}  95% CI [{result['lower']:.3f}, {result['upper']:.3f}]")

# Murphy decomposition — distinguishes miscalibration from discrimination loss
murphy = murphy_decomposition(y=actual, y_hat=predicted, exposure=exposure, distribution="poisson")
print(f"DSC (discrimination score): {murphy.discrimination:.4f}")
print(f"MCB (miscalibration):       {murphy.miscalibration:.4f}")
print(f"Verdict: {murphy.verdict}")   # 'RECALIBRATE' or 'REFIT'

PITMonitor — the standard pattern of running A/E tests monthly inflates the false-positive rate. After twelve months at α=0.05, the chance of a false alarm exceeds 40% even on a perfectly calibrated model. PITMonitor uses probability integral transform e-processes (Henzi, Murph, Ziegel 2025); the guarantee is P(ever raise an alarm | model calibrated) ≤ α, for all t, forever.

monitor = PITMonitor(alpha=0.05)
for row in live_data:
    mu = row.exposure * row.lambda_hat
    pit = float(poisson.cdf(row.claims, mu))
    alarm = monitor.update(pit)
    if alarm.triggered:
        print(f"Calibration alarm: evidence = {alarm.evidence:.2f}")
        break

Discrimination — Gini drift test

from insurance_monitoring import gini_drift_test_onesample, GiniDriftBootstrapTest

# One-sample design: reference data not required — just the stored training Gini
result = gini_drift_test_onesample(
    training_gini=0.42,
    monitor_actual=current_claims,
    monitor_predicted=current_predicted,
    monitor_exposure=current_exposure,
)
print(f"Gini change: {result.gini_change:+.3f}  p={result.p_value:.3f}  [{result.significant}]")

# Class-based API with governance plot (IFoA/PRA deliverable)
bt = GiniDriftBootstrapTest(training_gini=0.42, monitor_actual=current_claims,
                             monitor_predicted=current_predicted, monitor_exposure=current_exposure)
bt.test()
bt.plot()       # bootstrap histogram with CI shading
bt.summary()    # governance-ready paragraph

Sequential A/B testing — champion/challenger

Champion/challenger pricing experiments are routinely run with a pre-specified end date. Checking early to stop a bad challenger is statistically invalid under classical hypothesis testing. SequentialTest uses mixture SPRT (mSPRT, Johari et al. 2022) — check at every interim update with full type I error control.

from insurance_monitoring import SequentialTest

test = SequentialTest(metric="frequency", alpha=0.05)
for batch in monthly_batches:
    result = test.update(
        champion_claims=batch.champion_claims,
        champion_exposure=batch.champion_exposure,
        challenger_claims=batch.challenger_claims,
        challenger_exposure=batch.challenger_exposure,
    )
    print(f"e-value: {result.lambda_value:.2f}  stopped: {result.should_stop}")
    if result.should_stop:
        break

TRIPODD drift attribution

PSI tells you that driver_age has drifted. TRIPODD tells you whether that drift explains why the model's discrimination has fallen — accounting for feature interactions.

from insurance_monitoring import InterpretableDriftDetector

detector = InterpretableDriftDetector(
    error_control="fdr",        # Benjamini-Hochberg for d >= 10 factors
    loss="poisson_deviance",    # canonical for frequency models
)
detector.fit_reference(X_ref, y_ref, weights=exposure_ref)
result = detector.test(X_cur, y_cur, weights=exposure_cur)
print(result.significant_features)  # which factors explain the performance shift

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ScoreDecompositionTest — asymptotic inference on Murphy decomposition components

ScoreDecompositionTest decomposes a scoring rule into miscalibration (MCB), discrimination (DSC), and uncertainty (UNC) components with HAC standard errors, so you can formally test whether a model is miscalibrated vs. losing discrimination power — the distinction that drives the RECALIBRATE vs. REFIT decision.

Based on Dimitriadis & Puke (arXiv:2603.04275). Supports MSE, MAE, and quantile scoring rules. For two competing forecasts, the two-sample test has higher power than a plain Diebold-Mariano test when models differ in only one dimension.

import numpy as np
from insurance_monitoring.calibration import ScoreDecompositionTest

rng = np.random.default_rng(42)
n = 5_000

# Synthetic motor frequency: well-discriminating model with a calibration bias
y_hat = rng.gamma(2, 0.05, n)
exposure = rng.uniform(0.5, 1.5, n)
y = rng.poisson(exposure * y_hat * 1.08) / exposure  # 8% bias

sdi = ScoreDecompositionTest(score_type="mse")
result = sdi.fit_single(y, y_hat)
print(result.summary())
# MCB=0 test:   FAIL (miscalibrated)  p=0.0001
# DSC=0 test:   PASS (has skill)      p=0.0000

# Compare champion vs challenger on the same holdout
result_two = sdi.fit_two(y, y_hat, y_hat * 0.95)
print(result_two.summary())

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Regulatory context

PRA SoP3/24 requires an annual IMOR attestation from PRA-regulated insurers covering model governance, validation, and monitoring. The outputs of ModelMonitor and MonitoringReport are structured to feed directly into IMOR evidence packs: documented thresholds, RAG status, and a governance-ready summary paragraph for each model.

Note: PRA SS1/23 (Model Risk Management Principles for Banks) is a banking supervisory statement — it does not apply directly to Solvency II insurers. The mandatory monitoring obligations for GI pricing teams are SoP3/24 (PRA side) and Consumer Duty PS22/9 (FCA side). Many insurer MRM frameworks reference SS1/23 as useful practice guidance; this library is structured to meet those conventions where they add value.

Consumer Duty (PS22/9) requires ongoing monitoring of whether pricing outcomes are fair across customer groups. The combination of MonitoringReport and insurance-fairness calibration_by_group() produces a per-protected-group A/E split suitable for Consumer Duty Outcome 4 monitoring.

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Expected performance

On a 50,000-policy synthetic UK motor portfolio:

Task	Time	Notes
PSI on one feature	< 0.1s	Exposure-weighted
CSI across 10 features	< 0.5s	Returns Polars DataFrame
ae_ratio_ci()	< 0.1s	Wilson CI
MonitoringReport (no Murphy)	< 2s	A/E + Gini + CSI
MonitoringReport with Murphy	< 5s	Adds MCB/DSC decomposition
GiniDriftBootstrapTest (n_bootstrap=500)	5–15s	Percentile CI
SequentialTest batch update	< 0.1s	Per monthly update
InterpretableDriftDetector (10 features)	30–90s	FDR-controlled bootstrap
ModelMonitor.fit() (n_bootstrap=500, n=10k)	5–15s	Bootstrap Gini SE estimation
ModelMonitor.test() (n_bootstrap=500, n=5k)	5–15s	Gini + GMCB + LMCB bootstrap tests

Run the full workflow on Databricks

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References

Regulatory instruments

• PRA. (2024). Statement of Policy: Expectations for Insurers' Actuarial Function (SoP3/24). Prudential Regulation Authority. www.bankofengland.co.uk/prudential-regulation/publication/2024
• FCA. (2023). Consumer Duty: Final rules and guidance (PS22/9). Financial Conduct Authority. www.fca.org.uk/publications/policy-statements/ps22-9-new-consumer-duty

Population Stability Index and score monitoring

• Yurdakul, B. (2018). "Statistical Properties of Population Stability Index." University of KwaZulu-Natal working paper. (Derivation of PSI thresholds: <0.10 no change, 0.10–0.25 moderate shift, >0.25 major shift.)
• Hand, D.J. & Anagnostopoulos, C. (2014). "When is the area under the receiver operating characteristic curve an appropriate measure of classifier performance?" Pattern Recognition Letters, 42, 128–132. doi:10.1016/j.patrec.2014.01.015

Forecast scoring and Murphy decomposition

• Murphy, A.H. (1973). "A New Vector Partition of the Probability Score." Journal of Applied Meteorology, 12(4), 595–600. doi:10.1175/1520-0450(1973)012
• Gneiting, T. & Raftery, A.E. (2007). "Strictly Proper Scoring Rules, Prediction, and Estimation." Journal of the American Statistical Association, 102(477), 359–378. doi:10.1198/016214506000001437

Sequential testing and drift detection

• Page, E.S. (1954). "Continuous Inspection Schemes." Biometrika, 41(1/2), 100–115. doi:10.2307/2333009 (CUSUM control charts.)
• Wald, A. (1945). "Sequential Tests of Statistical Hypotheses." The Annals of Mathematical Statistics, 16(2), 117–186. doi:10.1214/aoms/1177731118

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Related libraries

Library	What it does
insurance-fairness	Per-protected-group A/E calibration, proxy detection, Consumer Duty audit reports
insurance-causal-policy	SDID causal inference — establishes whether a rate change caused an outcome shift
insurance-governance	Model risk committee packs and FCA Consumer Duty documentation
insurance-gam	Interpretable GLM-style models whose factors can be monitored directly

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Part of the Burning Cost open-source insurance analytics toolkit. → See all libraries

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