insurance-monitoring 0.7.0

Model drift detection, monitoring, calibration, and sequential A/B testing for insurance pricing models. PSI, CSI, Gini drift, A/E ratios, balance property testing, Murphy decomposition, TRIPODD drift attribution, mSPRT champion/challenger, bootstrap Gini CIs, PITMonitor e-process calibration monitoring.

insurance-monitoring

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Your aggregate A/E ratio looks fine. Your model has been mispricing under-25s for eight months.

Deployed insurance pricing models go stale. The portfolio ages, the claims environment shifts, regulators change the rules. Without systematic monitoring you find out about it when the loss ratio deteriorates — typically 12 to 18 months after the model started misfiring.

The central problem with aggregate A/E is that errors cancel at portfolio level. The model may be 15% cheap on young drivers and 15% expensive on mature drivers; the aggregate reads 1.00 and nobody raises an alarm. This library monitors the features, not just the headline number.

Why bother

Benchmarked on synthetic UK motor data — 50,000 training policies (2019–2021), monitored against a 2023 portfolio with known induced shifts: young drivers (under 25) oversampled 2x, high-risk areas (E and F) oversampled 50%, conviction points shifted upward for 20% of policies.

Note on benchmark sizes: the table below uses the 50,000/15,000 policy scenario. The detailed Performance section further down uses a smaller scenario (10,000 reference / 4,000 monitoring) for a faster local run. Both use the same DGP; the smaller run is provided for reproducibility without Databricks.

Monitoring check	Manual A/E check	MonitoringReport (PSI/CSI)	Notes
Aggregate A/E — shifted data	Computed	Same value	Both agree; neither alone is sufficient
driver_age distributional shift	Not detected	PSI RED (>0.25)	2x young driver oversampling
area distributional shift	Not detected	PSI AMBER/RED	High-risk area overweighting
conviction_points shift	Not detected	PSI AMBER	20% of policies shifted +1 point
Gini drift (ref vs shifted)	Not computed	Computed with bootstrap CI	Tests whether ranking has degraded
Structured audit trail	No	Yes (traffic-light report)	Suitable for inclusion in PRA SS1/23 model risk documentation (see note below)

The manual A/E check is blind to who is inside the portfolio. PSI per feature catches segment-level drift that cancels at portfolio level. The Gini drift z-test tells you whether the model's ranking has degraded — the difference between a cheap recalibration and a full refit.

Note on regulatory scope: PRA SS1/23 applies to banks and building societies. For insurers, the relevant standards are the PRA's internal model requirements under Solvency II and FCA Consumer Duty model governance obligations. The structured audit trail this library produces is appropriate for inclusion in model risk documentation under any of these frameworks.

▶ Run on Databricks

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Read more: Your Pricing Model is Drifting (and You Probably Can't Tell) — why PSI alone is insufficient, and what it means when A/E is stable but the Gini is falling.

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This library gives UK pricing teams two things in one install:

1. Ongoing model monitoring — exposure-weighted PSI for feature distribution, A/E ratios with Poisson confidence intervals, and the Gini drift z-test from arXiv 2510.04556.
2. Deep calibration diagnostics — balance property testing, auto-calibration, Murphy decomposition (UNC/DSC/MCB), and rectification methods for model sign-off and root-cause analysis (Lindholm & Wüthrich, SAJ 2025).

The two layers serve the same person — the pricing actuary — at different points in the model lifecycle. Use the monitoring layer for monthly/quarterly dashboards. Use the calibration suite when a model needs to be signed off or when monitoring flags a problem you need to diagnose.

No scikit-learn. No pandas. Polars-native throughout.

Installation

uv add insurance-monitoring

Quick example

This example uses named rating factors — which is how actuaries actually work with this data.

import polars as pl
import numpy as np
from insurance_monitoring import MonitoringReport

rng = np.random.default_rng(42)

# Reference period: training window
n_ref = 50_000
pred_ref = rng.uniform(0.05, 0.20, n_ref)
act_ref = rng.poisson(pred_ref).astype(float)

# Current monitoring period: 18 months into deployment
n_cur = 15_000
pred_cur = rng.uniform(0.05, 0.20, n_cur)
act_cur = rng.poisson(pred_cur * 1.08).astype(float)  # model underpredicted: actuals 8% above predictions (A/E ≈ 1.08)

# Feature DataFrames with named rating factors — pass these to get CSI per feature
feat_ref = pl.DataFrame({
    "driver_age":  rng.integers(18, 80, n_ref).tolist(),
    "vehicle_age": rng.integers(0, 15, n_ref).tolist(),
    "ncd_years":   rng.integers(0, 9, n_ref).tolist(),
})
feat_cur = pl.DataFrame({
    "driver_age":  rng.integers(25, 85, n_cur).tolist(),  # older drivers entering book
    "vehicle_age": rng.integers(0, 15, n_cur).tolist(),
    "ncd_years":   rng.integers(0, 9, n_cur).tolist(),
})

report = MonitoringReport(
    reference_actual=act_ref,
    reference_predicted=pred_ref,
    current_actual=act_cur,
    current_predicted=pred_cur,
    feature_df_reference=feat_ref,
    feature_df_current=feat_cur,
    features=["driver_age", "vehicle_age", "ncd_years"],
    murphy_distribution="poisson",
)

print(report.recommendation)
# 'RECALIBRATE' | 'REFIT' | 'NO_ACTION' | 'INVESTIGATE' | 'MONITOR_CLOSELY'

df = report.to_polars()
# metric              | value  | band
# ae_ratio            | 1.08   | amber
# gini_current        | 0.39   | amber
# gini_p_value        | 0.054  | amber
# csi_driver_age      | 0.14   | amber
# murphy_discrimination | 0.041 | RECALIBRATE
# murphy_miscalibration | 0.003 | RECALIBRATE
# recommendation      | nan    | RECALIBRATE

If you just want to run a quick sanity check without feature data:

import numpy as np
from insurance_monitoring import MonitoringReport

rng = np.random.default_rng(42)
pred_ref = rng.uniform(0.05, 0.20, 50_000)
act_ref = rng.poisson(pred_ref).astype(float)
pred_cur = rng.uniform(0.05, 0.20, 15_000)
act_cur = rng.poisson(pred_cur * 1.08).astype(float)

report = MonitoringReport(
    reference_actual=act_ref,
    reference_predicted=pred_ref,
    current_actual=act_cur,
    current_predicted=pred_cur,
    murphy_distribution="poisson",
)
print(report.recommendation)

Worked Example

model_drift_monitoring.py demonstrates the full monitoring stack on a synthetic motor book with three deliberately induced failure modes: covariate shift (older driver mix), calibration deterioration (segment-level A/E drift), and discriminatory power loss (Gini decay). It covers exposure-weighted PSI and CSI, segment A/E ratios with Poisson confidence intervals, the Gini drift z-test, and structured governance reporting suitable for inclusion in PRA SS1/23 model risk documentation.

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

Modules

calibration — A/E ratio, calibration suite, Murphy decomposition

The calibration module has two layers. Use A/E for routine monitoring. Use the calibration suite for model sign-off.

A/E ratio monitoring:

from insurance_monitoring.calibration import ae_ratio, ae_ratio_ci

# Aggregate A/E with Poisson CI (exact Garwood intervals)
result = ae_ratio_ci(actual, predicted, exposure=exposure)
# {'ae': 1.08, 'lower': 1.04, 'upper': 1.12, 'n_claims': 342, 'n_expected': 317}

# Segmented A/E: where is the model misfiring?
seg_ae = ae_ratio(
    actual, predicted, exposure=exposure,
    segments=driver_age_bands,
)
# Returns Polars DataFrame: segment | actual | expected | ae_ratio | n_policies

Calibration suite — model sign-off:

from insurance_monitoring.calibration import CalibrationChecker

checker = CalibrationChecker(distribution='poisson', alpha=0.05)
report = checker.check(y_holdout, y_hat_holdout, exposure_holdout)

print(report.verdict())    # 'OK' | 'RECALIBRATE' | 'REFIT'
print(report.summary())    # human-readable diagnostic paragraph

# Individual components
print(report.balance)          # BalanceResult: global A/E ratio with bootstrap CI
print(report.auto_calibration) # AutoCalibResult: per-cohort bootstrap MCB test
print(report.murphy)           # MurphyResult: UNC/DSC/MCB/GMCB/LMCB decomposition

Murphy decomposition directly:

from insurance_monitoring.calibration import murphy_decomposition

result = murphy_decomposition(y, y_hat, exposure, distribution='poisson')
# result.uncertainty     # baseline deviance (data difficulty)
# result.discrimination  # DSC: skill from ranking
# result.miscalibration  # MCB: excess from wrong price levels
# result.global_mcb      # GMCB: portion fixed by multiplying all predictions by A/E
# result.local_mcb       # LMCB: portion requiring model refit
# result.verdict         # 'OK' | 'RECALIBRATE' | 'REFIT'

Why two calibration layers? The A/E ratio answers "is the model globally right?". The Murphy decomposition answers "if it is wrong, is it wrong in a cheap way (scale factor) or an expensive way (the ranking is broken)?". You need both to make the RECALIBRATE vs REFIT decision correctly.

On the IBNR problem: the A/E ratio and balance test are only reliable on mature accident periods. For motor, at least 12 months of claims development. For liability, 24+ months. Apply chain-ladder factors first when monitoring recent accident months.

drift — Feature distribution monitoring

from insurance_monitoring.drift import psi, csi, ks_test, wasserstein_distance
import polars as pl

# PSI with exposure weighting (insurance-correct)
score_psi = psi(
    reference=score_train,
    current=score_q1_2025,
    n_bins=10,
    exposure_weights=earned_exposure,  # car-years, not policy count
)

# CSI heatmap across all rating factors
feature_ref = pl.DataFrame({"driver_age": [...], "vehicle_age": [...], "ncd_years": [...]})
feature_cur = pl.DataFrame({"driver_age": [...], "vehicle_age": [...], "ncd_years": [...]})
csi_table = csi(feature_ref, feature_cur, features=["driver_age", "vehicle_age", "ncd_years"])
# Returns: feature | csi | band

# Wasserstein: report drift in original units
d = wasserstein_distance(driver_ages_train, driver_ages_q1_2025)
print(f"Average driver age shifted by {d:.1f} years")

On exposure-weighted PSI: standard PSI treats every policy equally regardless of how long it was on risk. If your book renews quarterly and mixes 1-month and 12-month policies, unweighted PSI is wrong. The exposure_weights parameter weights bin proportions by earned exposure.

discrimination — Gini drift test

from insurance_monitoring.discrimination import gini_coefficient, gini_drift_test

gini_ref = gini_coefficient(act_ref, pred_ref, exposure=exp_ref)
gini_cur = gini_coefficient(act_cur, pred_cur, exposure=exp_cur)

# Statistical test: has Gini degraded significantly?
# Implements arXiv 2510.04556 Theorem 1
result = gini_drift_test(
    reference_gini=gini_ref,
    current_gini=gini_cur,
    n_reference=50_000,
    n_current=15_000,
    reference_actual=act_ref, reference_predicted=pred_ref,
    current_actual=act_cur, current_predicted=pred_cur,
)
# {'z_statistic': -1.93, 'p_value': 0.054, 'gini_change': -0.03, 'significant': False}

The Gini drift test is the distinguishing feature of this library. Most monitoring tools tell you whether A/E has moved. This tells you whether the model's ranking has degraded — the difference between a cheap recalibration and a full refit.

sequential — Anytime-valid champion/challenger testing (v0.5.0)

Standard A/B tests have a dirty secret: if you peek at results before the pre-specified end date and stop early when the data looks good, your actual false positive rate is far higher than your nominal 5%. On a monthly checking cadence, a fixed-horizon t-test can inflate to 25% FPR — five times nominal.

The sequential module implements the mixture Sequential Probability Ratio Test (mSPRT) from Johari et al. (2022). The test statistic is an e-process, which satisfies P(exists n: Lambda_n >= 1/alpha) <= alpha at all stopping times. You can check it every month for two years without inflating type I error. When it crosses the threshold, stop — the evidence is genuine.

Supports three metrics: claim frequency (Poisson rate ratio), claim severity (log-normal ratio), and combined loss ratio (product of the two e-values).

import datetime
from insurance_monitoring.sequential import SequentialTest

# Champion: existing model. Challenger: new model being tested.
# Feed monthly increments as they arrive — no need to wait for a fixed end date.

test = SequentialTest(
    metric="frequency",      # 'frequency' | 'severity' | 'loss_ratio'
    alternative="two_sided", # 'two_sided' | 'greater' | 'less'
    alpha=0.05,
    tau=0.03,                # prior std dev on log-rate-ratio: expect ~3% effects
    max_duration_years=2.0,
    min_exposure_per_arm=100.0,  # car-years before any stopping decision
)

# Month 1 — Q1 2025
result = test.update(
    champion_claims=42,   challenger_claims=38,
    champion_exposure=500, challenger_exposure=495,
    calendar_date=datetime.date(2025, 3, 31),
)
print(result.summary)
# "Challenger freq 8.8% lower (95% CS: 0.731–1.193). Evidence: 0.4 (threshold 20.0). Inconclusive."

# Month 4 — Q2 2025 (check as often as you like; FPR stays at 5%)
result = test.update(
    champion_claims=44,   challenger_claims=29,
    champion_exposure=510, challenger_exposure=505,
    calendar_date=datetime.date(2025, 6, 30),
)
print(result.decision)     # 'inconclusive' | 'reject_H0' | 'futility' | 'max_duration_reached'
print(result.should_stop)  # True when decision != 'inconclusive'

# Full history as Polars DataFrame
df = test.history()
# period_index | calendar_date | lambda_value | log_lambda_value | champion_rate | ...

For batch processing from a DataFrame of monthly reporting periods:

import polars as pl
from insurance_monitoring.sequential import sequential_test_from_df

monthly_data = pl.DataFrame({
    "date":               ["2025-01-31", "2025-02-28", "2025-03-31", "2025-04-30"],
    "champ_claims":       [42, 38, 44, 41],
    "champ_exposure":     [500, 490, 510, 495],
    "chall_claims":       [38, 31, 29, 28],
    "chall_exposure":     [495, 488, 505, 492],
})

result = sequential_test_from_df(
    df=monthly_data,
    champion_claims_col="champ_claims",
    champion_exposure_col="champ_exposure",
    challenger_claims_col="chall_claims",
    challenger_exposure_col="chall_exposure",
    date_col="date",
    metric="frequency",
    alpha=0.05,
)
print(result.summary)

When to use: Any champion/challenger experiment where results are checked before the pre-specified end date — which is almost every experiment in practice. Renewal cycles, rate change pilots, telematics scoring experiments.

When NOT to use: When you have a hard commitment to a fixed sample size and will genuinely not look before it completes. In that case, a standard two-sample test is more powerful than mSPRT.

On the prior tau: tau=0.03 encodes a prior that meaningful effects are around 3% on the log-rate-ratio scale. For telematics experiments where you expect larger effects (10%+), increase to tau=0.10. For fine-tuning experiments where the effect is expected to be very small, decrease to tau=0.01.

report — Combined monitoring in one call

from insurance_monitoring import MonitoringReport

report = MonitoringReport(
    reference_actual=act_ref,
    reference_predicted=pred_ref,
    current_actual=act_cur,
    current_predicted=pred_cur,
    exposure=exposure_cur,
    reference_exposure=exposure_ref,
    feature_df_reference=feat_ref,  # Polars DataFrame
    feature_df_current=feat_cur,
    features=["driver_age", "vehicle_age", "ncd_years"],
    murphy_distribution="poisson",
)

print(report.recommendation)
# 'REFIT' | 'RECALIBRATE' | 'NO_ACTION' | 'INVESTIGATE' | 'MONITOR_CLOSELY'

df = report.to_polars()
# metric              | value  | band
# ae_ratio            | 1.08   | amber
# gini_current        | 0.39   | amber
# gini_p_value        | 0.054  | amber
# csi_driver_age      | 0.14   | amber
# murphy_discrimination | 0.041 | RECALIBRATE
# murphy_miscalibration | 0.003 | RECALIBRATE
# recommendation      | nan    | RECALIBRATE

thresholds — Configurable traffic lights

from insurance_monitoring.thresholds import MonitoringThresholds, PSIThresholds

# Tighten PSI thresholds for a large motor book with monthly monitoring
custom = MonitoringThresholds(
    psi=PSIThresholds(green_max=0.05, amber_max=0.15),
)
report = MonitoringReport(..., thresholds=custom)

Default thresholds follow industry convention (PSI: 0.1/0.25 from FICO/credit scoring; A/E: 0.95–1.05 green, 0.90–1.10 amber; Gini: p < 0.32 amber, p < 0.10 red per arXiv 2510.04556 recommendation).

Decision framework

The recommendation property implements the three-stage decision tree from arXiv 2510.04556, mapped to actuarial practice:

Signal	Recommendation	Action
No drift in any test	NO_ACTION	Continue, schedule next review
A/E red, Gini stable	RECALIBRATE	Update intercept/offset (hours of work)
Gini red	REFIT	Rebuild model on recent data (weeks of work)
Both red	INVESTIGATE	Manual review — check data quality first
Any amber	MONITOR_CLOSELY	Increase monitoring frequency

When murphy_distribution is set, the Murphy decomposition sharpens the RECALIBRATE vs REFIT distinction: if GMCB > LMCB (global shift dominates), RECALIBRATE; if LMCB >= GMCB (local structure is broken), REFIT.

Calibration plots

The calibration module includes matplotlib visualisations for model documentation:

from insurance_monitoring.calibration import (
    CalibrationChecker,
    plot_auto_calibration,
    plot_murphy,
    plot_calibration_report,
)

checker = CalibrationChecker(distribution='poisson')
report = checker.check(y, y_hat, exposure)

# Three-panel combined figure (auto-calibration + Murphy bar + per-bin heatmap)
fig = plot_calibration_report(report)
fig.savefig("model_calibration_sign_off.pdf")

Databricks integration

The demo notebook at notebooks/demo_monitoring.py shows the full workflow on synthetic motor data and runs on Databricks serverless. Upload it to your workspace and schedule it as a monthly job against your MLflow inference table.

Background

The monitoring framework implements:

"Model Monitoring: A General Framework with an Application to Non-life Insurance Pricing", arXiv 2510.04556 (December 2025)

The calibration suite implements:

Lindholm & Wüthrich: "Three calibration properties for insurance pricing models" (SAJ 2025) Brauer et al.: arXiv:2510.04556 Section 4 — Murphy decomposition and the MCB bootstrap test

The sequential testing module implements:

Johari et al. (2022). "Always Valid Inference: Continuous Monitoring of A/B Tests." Operations Research 70(3). arXiv:1512.04922. Howard et al. (2021). "Time-uniform, nonparametric, nonasymptotic confidence sequences." Annals of Statistics 49(2).

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Capabilities Demo

Demonstrated on synthetic UK motor data with three deliberately induced failure modes: covariate shift (older drivers enter the book), calibration deterioration (claim frequency inflated for a segment), and stale discrimination (model trained on old data, portfolio composition changed). Full notebook: notebooks/benchmark.py.

• PSI/CSI flags the covariate shift — feature distributions in the monitoring period diverge from training, triggering configurable traffic lights (PSI > 0.25 = red)
• A/E ratio with confidence intervals catches calibration drift — segment-level actual-to-expected ratios with statistical significance tests, not just point estimates
• Gini drift z-test (arXiv 2510.04556) detects discrimination loss — the discriminatory power of the model has declined, which a standard A/E dashboard would miss
• MonitoringReport assembles all three checks into a single traffic-light summary with a recommended action: monitor, investigate, or refit

When to use: Any time more than a month has passed since the last model refit. A typical UK motor pricing cycle is 6–12 months between refits; covariate shift and calibration drift accumulate silently in between. Run the monitoring report monthly on the live book.

Databricks Notebook

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

Related libraries

Library	Why it's relevant
shap-relativities	Extract rating relativities from GBMs — when monitoring flags REFIT, use SHAP to diagnose which factors have drifted most
insurance-interactions	GLM interaction detection — a refit triggered by Gini degradation may need new interactions added
insurance-causal-policy	SDID causal evaluation — if monitoring shows deterioration after a rate change, use this to isolate cause
insurance-cv	Walk-forward cross-validation — use monitoring outputs to decide when to retrain and validate the retrained model
insurance-optimise	Constrained rate change optimisation — monitoring informs when a rate adjustment is needed; rate-optimiser determines the right one

All Burning Cost libraries →

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Performance

MonitoringReport vs manual A/E

Benchmarked against a manual aggregate A/E ratio check on synthetic UK motor insurance data — 10,000 reference policies and 4,000 monitoring-period policies with three deliberately induced failure modes. Full script: benchmarks/run_benchmark.py.

Check	Manual A/E	MonitoringReport
Reference A/E	0.9624	0.9624
Monitoring A/E	0.9420	0.9420
Manual verdict	INVESTIGATE	REFIT
Covariate shift (driver_age PSI = 0.21)	Not detected	AMBER
Calibration drift (new vehicles)	Not detected	Detected (Murphy)
Discrimination decay (30% predictions randomised)	Not detected	REFIT
Gini change	Not computed	−0.012
Gini drift p-value	N/A	0.76 (n=4,000 — underpowered)
Murphy discrimination	Not computed	REFIT flag
Murphy local MCB	Not computed	0.0090 (REFIT)

The aggregate A/E at 0.9420 falls just outside the 0.95–1.05 green band (verdict: INVESTIGATE), but it is blind to which segment is causing the drift and why. MonitoringReport identifies all three failure modes:

1. Covariate shift: driver_age PSI = 0.21 (AMBER). Young drivers (18–30) are oversampled 2x in the monitoring period.
2. Discrimination decay: Murphy decomposition flags REFIT — the local MCB (0.0090) exceeds global MCB (0.0002), meaning the model's ranking is broken, not just the scale.
3. Calibration drift: detected via the Murphy miscalibration component.

The Gini drift test returns p=0.76 at n=4,000, which is correct — 4,000 policies does not give enough statistical power to detect a Gini drop of −0.012. At 15,000 policies the same DGP produces z≈−1.9, p≈0.06. The test is appropriately conservative at small sample sizes.

When to use: Any time more than a month has passed since the last model refit. The monitoring report runs in under 40 seconds on 14,000 policies (including bootstrap variance estimation for the Gini test).

SequentialTest (mSPRT) vs fixed-horizon t-test

Benchmarked on simulated UK motor champion/challenger data. 10,000 Monte Carlo simulations under H0 (no true effect): analyst checks results monthly for 24 months and stops if the test is significant. Full script: benchmarks/benchmark_sequential.py.

Method	Nominal FPR	Actual FPR (monthly peeking)	Notes
Fixed-horizon t-test	5%	~25%	5x inflation from repeated peeking
mSPRT (SequentialTest)	5%	~1%	Valid at all stopping times

Under H1 (challenger 10% cheaper on frequency), mSPRT detects the effect in a median of 8 months on a 500-policy-per-arm-per-month book; a pre-registered t-test at 24 months would reach the same conclusion but forces the team to wait.

The 25% FPR figure for the fixed-horizon t-test assumes monthly checks from month 1 with early stopping on significance — the common practice of "we'll check again next month to see if it's still significant." If the analyst genuinely never looks before month 24, the t-test is valid; in practice, nobody does this.

Related Libraries

Library	What it does
insurance-deploy	Champion/challenger deployment — monitoring informs when to switch challenger to champion
insurance-cv	Walk-forward cross-validation — produces the baseline metrics that monitoring tracks prospectively
insurance-covariate-shift	Covariate shift detection and correction — use when monitoring flags PSI drift requiring model adaptation

Licence

BSD-3