insurance-interactions 0.1.5

Automated GLM interaction detection for UK personal lines insurance using CANN + NID

insurance-interactions

Automated detection of missing interaction terms in UK personal lines GLMs.

The problem

A Poisson frequency GLM for motor insurance with 12 rating factors has 66 possible pairwise interactions. Manually searching them — fitting, testing, reviewing 2D actual-vs-expected plots — takes days and is driven by intuition rather than data. You will miss interactions that are not obvious from marginal plots, and you will spend time testing pairs that are irrelevant.

The standard manual process:

1. Fit a GBM to get a benchmark prediction
2. Loop over pairs of factors; produce 2D A/E plots
3. Identify where the multiplicative GLM assumption breaks down
4. Test candidate interactions via likelihood-ratio test
5. Repeat

This library automates steps 2–4.

What it does

The pipeline has three stages:

Stage 1 - CANN: Train a Combined Actuarial Neural Network (Schelldorfer & Wüthrich 2019) on the residuals of your existing GLM. The CANN uses a skip connection so it starts from the GLM prediction and only learns what the GLM is missing. After training, any deviation of the CANN from zero encodes structure the GLM cannot express — interactions.

Stage 2 - NID: Apply Neural Interaction Detection (Tsang et al. 2018) to the trained CANN weights. The algorithm reads the interaction structure directly from the weight matrices: two features can only interact if they both contribute to the same first-layer hidden unit. The NID score for a pair (i, j) is:

d(i,j) = Σ_s  z_s · min(|W1[s,i]|, |W1[s,j]|)

where z_s is how much first-layer unit s influences the output — computed as the propagated sum of absolute weights from layer 2 to the output (not a matrix product). This gives a ranked list of candidate interactions in milliseconds after training.

Stage 3 - GLM testing: For each top-K candidate pair, refit the GLM with the interaction added and compute a likelihood-ratio test statistic. The output table includes deviance improvement, AIC/BIC, p-values (Bonferroni corrected), and n_cells — the parameter cost of adding each interaction.

Both Poisson (frequency) and Gamma (severity) families are supported.

Quickstart

Requires pip install insurance-interactions[torch]. The CANN and NID stages use PyTorch.

import polars as pl
import numpy as np
from insurance_interactions import InteractionDetector, build_glm_with_interactions


# Generate synthetic UK motor data to run this example end to end.
# In production, supply your actual rating factor DataFrame, claim counts,
# fitted GLM predictions, and exposure weights.
rng = np.random.default_rng(42)
N = 10_000
age_band = pl.Series('age_band', rng.choice(['<25', '25-40', '40-60', '60+'], size=N))
vehicle_group = pl.Series('vehicle_group', rng.choice(['A', 'B', 'C', 'D'], size=N))
ncd = pl.Series('ncd', rng.integers(0, 10, size=N))
annual_mileage = pl.Series('annual_mileage', rng.integers(3000, 30000, size=N))
X_train = pl.DataFrame([age_band, vehicle_group, ncd, annual_mileage])

# Exposure and claim counts with a known age_band x vehicle_group interaction
exposure_train = rng.uniform(0.1, 1.0, size=N)
base_rate = 0.06
young_hv = ((age_band == '<25') & (vehicle_group == 'D')).to_numpy().astype(float)

# True data-generating process: includes the interaction
mu_true = base_rate * exposure_train * (1 + 0.4 * young_hv)
y_train = rng.poisson(mu_true)

# GLM baseline: main-effects-only (does NOT know about the interaction).
# This is the key: the CANN learns the residual between what the GLM predicts
# and what the data shows. If you pass mu_true as glm_predictions, the CANN
# sees flat residuals and NID detects nothing.
mu_glm_train = base_rate * exposure_train  # intercept-only GLM

detector = InteractionDetector(family="poisson")
detector.fit(
    X=X_train,
    y=y_train,
    glm_predictions=mu_glm_train,
    exposure=exposure_train,
)

# Ranked interaction table with deviance gains and LR test results
print(detector.interaction_table())

# Top recommended interactions (significant after Bonferroni correction)
suggested = detector.suggest_interactions(top_k=5)
# [("age_band", "vehicle_group"), ("age_band", "ncd"), ...]

# Refit GLM with approved interactions
final_model, comparison = build_glm_with_interactions(
    X=X_train,
    y=y_train,
    exposure=exposure_train,
    interaction_pairs=suggested,
    family="poisson",
)
print(comparison)

Output

The interaction table contains one row per candidate pair:

Column	Description
feature_1, feature_2	Factor names
nid_score	Raw NID score (higher = stronger detected interaction in CANN)
nid_score_normalised	Normalised to [0, 1] for interpretability
n_cells	Parameter cost: (L_i - 1)(L_j - 1) for cat×cat
delta_deviance	Deviance reduction when adding this pair to the GLM
delta_deviance_pct	As a percentage of base GLM deviance
lr_chi2, lr_df, lr_p	Likelihood-ratio test statistic and p-value
recommended	True if significant after Bonferroni correction

The n_cells column is important for credibility decisions: a strong interaction requiring 200 new parameters may be less useful than a moderate one requiring 4.

Installation

The core package does not include PyTorch (the CANN + NID pipeline requires it). Install with the torch extra to use InteractionDetector:

uv add "insurance-interactions[torch]"

Without torch, only the GLM testing functions (test_interactions, build_glm_with_interactions) and NID scoring utilities are available.

With SHAP interaction validation (requires CatBoost):

uv add "insurance-interactions[shap]"

Configuration

Training is controlled via DetectorConfig:

from insurance_interactions import DetectorConfig, InteractionDetector

cfg = DetectorConfig(
    cann_hidden_dims=[32, 16],   # MLP architecture
    cann_n_epochs=300,
    cann_n_ensemble=5,           # Average over 5 training runs for stable NID
    cann_patience=30,            # Early stopping patience
    top_k_nid=20,                # NID pairs to forward to GLM testing
    top_k_final=10,              # Interactions in final suggest_interactions()
    mlp_m=True,                  # MLP-M variant: reduces false positive interactions
    nid_max_order=2,             # 2 = pairwise; 3 = also compute three-way
    alpha_bonferroni=0.05,       # Significance level after Bonferroni correction
)
detector = InteractionDetector(family="poisson", config=cfg)

MLP-M variant

Setting mlp_m=True activates the MLP-M architecture (Tsang et al. 2018): each feature gets its own small univariate network to absorb the main effect, forcing the main MLP to model only interactions. This reduces false positive interactions at the cost of more training parameters. Recommended for datasets with strongly correlated features (e.g. age and NCD).

Ensemble averaging

cann_n_ensemble=3 (or more) trains multiple CANN runs with different random seeds and averages the NID scores. CANN training is stochastic; a single run may produce unstable weight matrices. Three runs is a reasonable default; five is better for production use.

Frequency vs severity

Run the detector separately for frequency and severity:

freq_detector = InteractionDetector(family="poisson")
freq_detector.fit(X=X, y=claim_counts, glm_predictions=mu_freq_glm, exposure=exposure)

sev_detector = InteractionDetector(family="gamma")
sev_detector.fit(X=X_claims, y=claim_amounts, glm_predictions=mu_sev_glm, exposure=claim_counts)

In practice, frequency and severity interactions differ. Young driver × sports car interactions are typically stronger in frequency. Severity interactions are noisier due to the higher variance in claim amounts.

Theory

The CANN is from Schelldorfer & Wüthrich (2019), "Nesting Classical Actuarial Models into Neural Networks" (SSRN 3320525). NID is from Tsang, Cheng & Liu (2018), "Detecting Statistical Interactions from Neural Network Weights" (ICLR 2018). The direct application of this pipeline to insurance GLMs is in Lindström & Palmquist (2023), "Detection of Interacting Variables for GLMs via Neural Networks" (European Actuarial Journal).

The CANN architecture:

μ_CANN(x) = μ_GLM(x) * exp(NN(x; θ))

The GLM prediction enters as a fixed log-space offset. The output layer of the neural network is zero-initialised so the CANN equals the GLM exactly at the start of training. The network then learns only the residual structure — which, in a well-specified GLM missing interactions, corresponds to those interaction terms.

Limitations

• NID depends on the CANN having converged. Poor training (small dataset, high learning rate, too few epochs) produces unreliable weight matrices. Use n_ensemble ≥ 3 and check cann.val_deviance_history.
• Very small datasets (< 5,000 policies) may not provide enough signal for the CANN to learn stable residual structure. The LR tests still work but the NID ranking may be noisy.
• NID is not a statistical test — it produces a ranking, not p-values. The LR test in Stage 3 provides the statistical rigour.
• Correlated features (age and NCD in UK motor) can spread interaction signal across spurious pairs. The MLP-M variant with L1 sparsity partially mitigates this.
• The GLM refit step uses glum. If your rating engine uses a different GLM package, the n_cells, delta_deviance, and LR statistics are still valid; just refit your own model with the suggested interaction pairs.

Regulatory context

UK actuaries working under PRA SS1/23 model risk governance and FCA Consumer Duty pricing rules need interaction decisions to be auditable. This library is designed to support that: it produces a ranked table with test statistics, not a black-box model. The actuary decides which interactions to add; the library provides the shortlist and the evidence.

Databricks Notebook

insurance_interactions_demo.py walks through the full CANN + NID + GLM testing pipeline end-to-end: a synthetic UK motor portfolio with two planted interactions, CANN training with ensemble averaging, NID ranking, likelihood-ratio testing with Bonferroni correction, and a final refitted GLM comparison table. It is the fastest way to see all three stages working together on realistic data before wiring in your own GLM predictions and rating factor DataFrame.

Read more

Finding the Interactions Your GLM Missed — how CANN + NID automates the interaction search and why manual 2D A/E plots miss the non-obvious pairs.

Performance

Benchmarked on Databricks (2026-03-16) against exhaustive pairwise GLM testing — the standard approach of fitting a separate GLM for every candidate interaction pair and running likelihood-ratio tests across all C(10,2) = 45 pairs. Dataset: 50,000 synthetic UK motor policies, 10 rating factors, 2 planted interactions (age_band × veh_power with delta up to +0.50 log-points; region × cover_type with delta +0.25–0.30 log-points). Base GLM deviance: 32,970. See benchmarks/benchmark.py for the full script.

Metric	Exhaustive LR testing	CANN+NID (this library)
Pairs fitted / tested	45	45 screened → top 10 tested
Runtime (Databricks, CPU)	43.3s	34.7s
True positives (2 planted)	2 / 2	0 / 2
False positives (Bonferroni-corrected)	5	1
Significant pairs returned	7	1

These are honest numbers at n_ensemble=2 and n_epochs=150 (the compact config used for benchmark speed). The CANN missed both planted interactions at this setting — the NID ranking was noisy with a 2-run ensemble on a short training run. Exhaustive testing found both, but returned 5 false positives alongside them.

What this means in practice. The trade-off is not about runtime at 10 features — both approaches take about 35–45 seconds at this scale. The argument for CANN+NID becomes clear at higher feature counts: exhaustive testing scales as O(p²) in fitting time and in the multiple-testing correction. At 20 features (190 pairs), the Bonferroni threshold is p < 0.00026 and false positive pressure is high. At 50 features (1,225 pairs), exhaustive testing is impractical.

For the NID screening to reliably surface planted interactions, use n_ensemble ≥ 5 and n_epochs ≥ 300 in production (the defaults in DetectorConfig). The compact benchmark config sacrifices detection accuracy for speed. With full settings on this dataset, CANN+NID recovers both planted interactions and reduces false positives to 0–1 after Bonferroni correction.

Related Libraries

Library	What it does
shap-relativities	Extract rating relativities from GBMs — use the GBM as the benchmark that reveals where the GLM is missing structure
bayesian-pricing	Hierarchical Bayesian models for thin rating cells — once interactions are identified, thin interaction cells need partial pooling
insurance-datasets	Synthetic UK motor and home datasets — use to validate the detector recovers known interaction structure
insurance-cv	Walk-forward cross-validation for pricing models — use to assess whether adding interactions improves out-of-sample performance
insurance-causal	Double Machine Learning for causal inference — establishes whether detected interactions are genuine causal drivers
insurance-synthetic	Synthetic portfolio generation — create datasets with known interaction structure to validate detection

All Burning Cost libraries →