insurance-pin

Tree-like Pairwise Interaction Networks for insurance pricing.

PIN is a neural GA2M — the same additive structure as a GLM with interaction terms, but each shape function is parameterised by a shared neural network keyed by learned interaction tokens. One small network. All pairs. Best published result on French MTPL at 4,147 parameters.

The Problem

Pricing teams need models that:

1. Produce predictions that decompose — so you can explain why risk A is rated differently from risk B, not just that it is.
2. Capture interactions between features — bonus-malus and vehicle brand are not independent, and pretending they are leaves money on the table.
3. Beat GAMs and GLM-with-interactions on held-out deviance — otherwise why bother with the added complexity?

EBMs (GA2M) solve 1 and 2 but are tree-based and stage-wise. FNNs solve 3 but fail 1 and 2. PIN solves all three.

The Model

f_PIN(x) = exp( sum_{j<=k} w_{jk} * h_{jk}(x) + b )

where:

h_{jk}(x) = clamp((1 + f_theta(phi_j(x_j), phi_k(x_k), e_{jk})) / 2, 0, 1)

• phi_j(x_j) — per-feature embedding (not shared). 2-layer FNN for continuous, entity embedding for categorical. Output dimension d.
• f_theta — one shared 3-layer FNN for all pairs. Input: [phi_j, phi_k, e_{jk}].
• e_{jk} — learned interaction token (d0-dimensional). This is the key idea: one network, pair-specific behaviour via tokens. Analogous to CLS tokens in BERT.
• w_{jk} — scalar output weight per pair.
• Diagonal terms (j=k) are main effects. Off-diagonal (j<k) are interactions.

Paper: Richman, Scognamiglio, Wüthrich. "Tree-like Pairwise Interaction Networks." arXiv:2508.15678 (August 2025).

Result on freMTPL2freq (Poisson deviance x10^-2):

Model	Deviance
Null model	25.445
Poisson GLM	24.102
Poisson GAM	23.956
Ensemble FNN	23.783
Ensemble CAFFT (27k params)	23.726
Ensemble Credibility TRM	23.711
Ensemble PIN (4,147 params)	23.667

Install

pip install insurance-pin

Requires: torch>=2.0, polars>=0.20, numpy>=1.24, matplotlib>=3.7.

Usage

from insurance_pin import PINModel, PINEnsemble, PINDiagnostics

# Feature specification
features = {
    "age_driver":  "continuous",
    "bonus_malus": "continuous",
    "density":     "continuous",
    "veh_age":     "continuous",
    "drive_age":   "continuous",
    "veh_power":   "continuous",
    "longitude":   "continuous",
    "veh_brand":   11,   # 11 categories
    "region":      22,   # 22 regions
}

# Reference config from paper: 4,147 parameters for 9 features
model = PINModel(
    features=features,
    embedding_dim=10,      # d
    hidden_dim=20,         # d' (continuous embedding hidden width)
    token_dim=10,          # d0
    shared_dims=(30, 20),  # d1, d2
    loss="poisson",        # "gamma" or "tweedie" also available
    lr=0.001,
    batch_size=128,
    max_epochs=300,
    patience=20,
)

# y is FREQUENCY (claims / exposure), not claim count
model.fit(X_train, y_train, exposure=exposure_train)

# Predict frequency
freq_pred = model.predict(X_test)

# Predict expected claims (frequency * exposure)
claims_pred = model.predict(X_test, exposure=exposure_test)

Ensemble (recommended for production)

ensemble = PINEnsemble(
    n_models=10,
    features=features,
    **same_kwargs_as_above,
)
ensemble.fit(X_train, y_train, exposure=exposure_train)

freq_pred = ensemble.predict(X_test)
uncertainty = ensemble.predict_std(X_test)  # epistemic uncertainty

Interpretability

diag = PINDiagnostics(model)

# Which pairs matter most?
diag.interaction_heatmap()
fig, ax, importance = diag.weighted_importance(X_background)

# Main effect curves
diag.plot_main_effect("bonus_malus", X_background)

# Interaction surfaces
diag.plot_surface("bonus_malus", "veh_brand", X_background)

# Exact SHAP — not an approximation
# Cost: 2*(q+1) forward passes per sample per background sample
shap_values = model.shapley_values(X_test, X_background, n_background=200)

Access raw pair contributions

# Returns w_{jk} * h_{jk}(x) for each pair, shape (n,)
contribs = model.pair_contributions(X_test)

# sum(contribs.values()) + bias ≈ log(prediction)  [linear predictor scale]

Architecture Details

Why shared weights?

A separate network per pair (like ANAM) would require O(q^2) networks. For q=9 features that's 45 networks. Instead, PIN trains one network f_theta and differentiates pairs via learned tokens e_{jk}. This is what keeps the param count at 4,147 while modelling all 45 pairs simultaneously.

Interaction tokens

Each pair (j,k) gets a learned d0-dimensional vector. These are nn.Parameters initialized near zero and trained alongside all other weights. The token tells f_theta which pair it's computing, analogously to how BERT's CLS token identifies the task.

Centered hard sigmoid

The activation is clamp((1+x)/2, 0, 1):

• x=0 maps to 0.5 (centred at origin)
• x=±1 are the saturation points
• Gradient is 0.5 in the linear region

This is not torch.nn.Hardsigmoid, which uses clamp((x+3)/6, 0, 1).

Post-hoc centering

After fitting, we subtract the training mean of each w_{jk} * h_{jk} term. The paper doesn't do this, but it's needed for production: without it, w_{jk} is not interpretable because the pair terms have non-zero means that all absorb into the bias term in a non-transparent way.

Exposure

Exposure enters as a multiplicative weight on the deviance bracket:

L = (1/n) sum 2 * v_i * (mu_i - Y_i - Y_i * log(mu_i / Y_i))

where v_i is exposure (years at risk) and Y_i = claims_i / v_i is frequency. This is the standard actuarial formulation, not a log offset.

Exact Shapley Values

Because PIN is pairwise additive, Shapley values are exact at cost 2(q+1) forward passes per sample per background sample. For q=9 and 200 background samples, decomposing 100 predictions takes ~20 seconds.

The values are on the linear predictor scale. Exponentiating gives multiplicative relativities, which is the natural language of insurance rating.

Comparison with Other Libraries

Library	Architecture	Use when
insurance-ebm	Cyclic boosted trees (GA2M)	You need auditable lookup tables
insurance-anam	Neural additive model with monotonicity	You need monotone constraints
insurance-interactions	Post-hoc interaction detection (NID)	You want to rank interactions in an existing model
insurance-pin	Shared-weight neural GA2M	You want best predictive performance with full decomposability

Supported Loss Functions

• poisson — Poisson deviance (frequency modelling)
• gamma — Gamma deviance (severity modelling)
• tweedie — Tweedie deviance, 1 < p < 2 (pure premium)

License

MIT