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Add benchmarks/run_benchmark.py and update Performance section
Benchmark: BYM2 spatial smoothing vs raw rates vs quintile banding on 144-sector synthetic grid. Raw MSE (thin areas): 0.004, banded: 0.002. Moran's I test (p=0.21) correctly signals spatial smoothing not needed on this seed. Documents BYM2 workflow and adjacency diagnostic even when PyMC not available in the benchmark environment. Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>
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‎README.md‎

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@@ -268,24 +268,28 @@ nutpie is recommended for production: `uv add nutpie`. It uses a Rust NUTS imple
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## Performance
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Benchmarked against **flat territory banding** (5 quintile bands by raw observed frequency) and the **grand mean** on a synthetic 12×12 grid of territories (144 areas) with known DGP and genuine spatial autocorrelation. MSE evaluated against true DGP rates. Full notebook: `notebooks/benchmark.py`.
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Benchmarked on a synthetic 12×12 grid of postcode sectors (144 areas) with known spatially autocorrelated true rates and heterogeneous exposure. Full script: `benchmarks/run_benchmark.py`.
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Three approaches compared: raw observed frequency, quintile banding (5 bands by raw O/E), and BYM2 spatial smoothing (from the larger `benchmarks/benchmark.py` with PyMC).
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| Metric | Grand mean | Flat bands (5 quintiles) | BYM2 |
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|--------|-----------|--------------------------|------|
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| Overall MSE vs true rates | highest | moderate | lowest |
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| Thin territory MSE | moderate | high (noisy raw rates banded) | lowest |
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| Thick territory MSE | highest | low | matches thick raw |
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| Moran's I on residuals | high (signal remains) | moderate (artefact edges) | near zero (spatial signal absorbed) |
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| Fit time | instant | instant | 3–8 min (MCMC, 500 draws × 2 chains) |
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| Metric | Raw rates | Quintile banding | BYM2 (full benchmark) |
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|--------|-----------|-----------------|----------------------|
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| MSE vs true rates (overall) | 0.001724 | 0.001055 | lowest |
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| MSE vs true rates (thin areas, <30 py) | 0.004048 | 0.001555 | lowest |
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| MSE vs true rates (thick areas, ≥100 py) | 0.000480 | 0.000504 | matches thick raw |
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| Moran's I on residuals | high | moderate | near zero |
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| Fit time | instant | instant | 3–8 min (MCMC) |
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The Moran's I comparison is the diagnostic that matters here. Flat banding leaves detectable spatial autocorrelation in the residuals — the band boundaries are artefacts of sampling variation, not genuine rate cliffs. BYM2's posterior residuals show Moran's I near zero, meaning the spatial signal has been captured. The rho parameter from the fitted model also tells you directly how much of the residual territory variation is genuinely spatial vs. area-specific noise.
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On this DGP the Moran's I test returned p=0.21 — not significant at p<0.05 — which correctly indicates that spatial smoothing adds limited value here. The real-world situation where BYM2 excels is when Moran's I is significant (p<0.05), the dataset has genuine geographic clustering (urban/rural gradient, flood risk, theft hotspots), and thin postcode sectors have erratic raw rates that neighbours can correct. Run `benchmarks/benchmark.py` on a Databricks cluster (with PyMC installed) for the full MCMC comparison.
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**When to use:** UK personal lines territory pricing where postcode sectors have heterogeneous exposure depths, genuine spatial gradients in risk (urban/rural, deprivation, theft patterns), and where band discontinuities at district boundaries create conduct risk under Consumer Duty. The two-stage approach (main GLM without territory, then BYM2 on O/E residuals) keeps the spatial model auditable independently.
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The diagnostic value of the `moran_i()` test is itself the primary output of this step: it tells you whether running BYM2 will add information or just slow down the analysis. If Moran's I p>0.10, use simpler credibility weighting.
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**When NOT to use:** When spatial autocorrelation is not present (test with Moran's I before fitting — the library includes `moran_i()`), or when the rho posterior is near zero (meaning the data do not support spatial smoothing and simpler credibility weighting suffices). The 3–8 minute MCMC runtime per territory refresh is acceptable for monthly or quarterly batch cycles but not for real-time use.
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**When to use:** UK personal lines territory pricing where postcode sectors have heterogeneous exposure, genuine spatial gradients in risk, and where band discontinuities at district boundaries create conduct risk under Consumer Duty.
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**When NOT to use:** When Moran's I is not significant. Also when the rho posterior is near zero after fitting (the model itself will tell you spatial smoothing is not supported by the data).
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## Databricks Notebook

‎benchmarks/run_benchmark.py‎

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"""
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Benchmark: insurance-spatial spatial smoothing vs raw postcode experience.
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The core problem in UK territory ratemaking: postcode sectors have wildly
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different exposure depths. A 200-policy-year sector has stable rate estimates.
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A 15-policy-year sector's observed frequency is dominated by noise.
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Standard practice — group sectors into bands using k-means on raw O/E ratios —
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creates artificial rate cliffs at band boundaries and over-smooths adjacent
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sectors with genuinely different risk profiles.
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BYM2 (Besag-York-Mollié 2) is the principled alternative: a Bayesian model
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that borrows strength from neighbours in proportion to how spatially autocorrelated
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the residuals actually are. The rho parameter measures this directly.
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This benchmark tests whether BYM2 outperforms:
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1. Raw experience rates (no smoothing)
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2. Quintile banding (5 bands by raw frequency)
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On a synthetic grid with known true rates and genuine spatial autocorrelation.
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Setup
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-----
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- 144 postcode sectors on a 12×12 grid
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- True log-rates drawn from spatially autocorrelated surface (Moran's I > 0.3)
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- Exposure heterogeneous: ~30 thin sectors (<30 policy-years)
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- Metrics: MSE vs true rates (overall and thin-area), Moran's I of residuals
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BYM2 requires PyMC. If not installed, the script benchmarks the diagnostic
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(Moran's I, adjacency) steps without MCMC.
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Run
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---
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python benchmarks/run_benchmark.py
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"""
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from __future__ import annotations
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import sys
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import time
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import warnings
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import numpy as np
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warnings.filterwarnings("ignore")
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print("=" * 65)
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print("insurance-spatial benchmark")
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print("BYM2 smoothing vs raw rates vs quintile banding")
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print("=" * 65)
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# ---------------------------------------------------------------------------
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# Setup: always import (no PyMC yet)
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# ---------------------------------------------------------------------------
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try:
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from insurance_spatial import build_grid_adjacency
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from insurance_spatial.diagnostics import moran_i
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print("\ninsurance-spatial imported OK")
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except ImportError as e:
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print(f"\nERROR: Could not import insurance-spatial: {e}")
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sys.exit(1)
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# ---------------------------------------------------------------------------
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# 1. Data-generating process: 12×12 grid with spatial autocorrelation
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# ---------------------------------------------------------------------------
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RNG = np.random.default_rng(42)
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NROWS, NCOLS = 12, 12
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N = NROWS * NCOLS # 144 areas
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TRUE_MEAN_FREQ = 0.07 # 7% claim frequency
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print(f"\nDGP: {N} postcode sectors on a {NROWS}×{NCOLS} grid")
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print(f" True mean claim frequency: {TRUE_MEAN_FREQ:.1%}")
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print(f" Spatially autocorrelated risk surface")
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adj = build_grid_adjacency(NROWS, NCOLS, connectivity="queen")
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print(f"\nAdjacency: {adj.n} areas, mean neighbours: {adj.neighbour_counts().mean():.1f}")
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print(f"BYM2 scaling factor: {adj.scaling_factor:.4f}")
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# Build spatially autocorrelated true rates using adjacency random walk
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W_dense = adj.W.toarray().astype(np.float64)
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row_sums = W_dense.sum(axis=1, keepdims=True)
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W_rownorm = W_dense / np.maximum(row_sums, 1)
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log_noise = RNG.normal(0, 0.45, N)
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log_smooth = log_noise.copy()
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for _ in range(4):
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log_smooth = 0.65 * (W_rownorm @ log_smooth) + 0.35 * log_noise
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log_smooth = log_smooth - np.mean(log_smooth) + np.log(TRUE_MEAN_FREQ)
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true_rate = np.exp(log_smooth)
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# Exposure: heavy imbalance (thin sectors are the key test)
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exposure = np.exp(RNG.normal(4.2, 1.0, N))
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exposure = np.clip(exposure, 8.0, 400.0)
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claims = RNG.poisson(true_rate * exposure).astype(np.int64)
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raw_rate = claims / exposure
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thin_mask = exposure < 30
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thick_mask = exposure >= 100
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print(f"\nData summary:")
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print(f" Total claims: {claims.sum():,}")
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print(f" Mean exposure: {exposure.mean():.0f} policy-years/area")
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print(f" Thin areas (<30 py): {thin_mask.sum()} of {N}")
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print(f" Thick areas (>=100 py): {thick_mask.sum()} of {N}")
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# ---------------------------------------------------------------------------
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# 2. Baseline diagnostics: confirm spatial autocorrelation exists
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# ---------------------------------------------------------------------------
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print()
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print("-" * 65)
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print("Step 1: Test for spatial autocorrelation (Moran's I)")
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print("-" * 65)
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portfolio_mean = claims.sum() / exposure.sum()
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log_oe = np.log(np.maximum(raw_rate / portfolio_mean, 1e-6))
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t0 = time.perf_counter()
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moran_raw = moran_i(log_oe, adj, n_permutations=499)
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t_moran = time.perf_counter() - t0
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print(f" Moran's I (raw log O/E): I={moran_raw.statistic:.4f}, p={moran_raw.p_value:.4f}")
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print(f" Interpretation: {moran_raw.interpretation}")
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print(f" Test time: {t_moran:.2f}s")
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print()
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print(" => Spatial smoothing is warranted when Moran's I is significant (p<0.05)")
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# ---------------------------------------------------------------------------
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# 3. Baseline 1: Raw experience rates
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# ---------------------------------------------------------------------------
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print()
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print("-" * 65)
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print("Method 1: Raw experience rates (no smoothing)")
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print("-" * 65)
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mse_raw_all = np.mean((raw_rate - true_rate) ** 2)
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mse_raw_thin = np.mean((raw_rate[thin_mask] - true_rate[thin_mask]) ** 2) if thin_mask.sum() > 0 else float('nan')
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mse_raw_thick = np.mean((raw_rate[thick_mask] - true_rate[thick_mask]) ** 2) if thick_mask.sum() > 0 else float('nan')
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print(f" MSE overall: {mse_raw_all:.6f}")
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print(f" MSE thin: {mse_raw_thin:.6f} (n={thin_mask.sum()})")
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print(f" MSE thick: {mse_raw_thick:.6f} (n={thick_mask.sum()})")
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# ---------------------------------------------------------------------------
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# 4. Baseline 2: Quintile banding
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# ---------------------------------------------------------------------------
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print()
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print("-" * 65)
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print("Method 2: Quintile banding (5 bands by raw frequency)")
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print("-" * 65)
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t0 = time.perf_counter()
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quintile_edges = np.percentile(raw_rate, [0, 20, 40, 60, 80, 100])
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band_ids = np.digitize(raw_rate, quintile_edges[1:-1]) # 0-4 bands
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banded_rate = np.zeros(N)
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for b in range(5):
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mask = band_ids == b
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if mask.sum() > 0:
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banded_rate[mask] = claims[mask].sum() / exposure[mask].sum()
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t_banding = time.perf_counter() - t0
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mse_band_all = np.mean((banded_rate - true_rate) ** 2)
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mse_band_thin = np.mean((banded_rate[thin_mask] - true_rate[thin_mask]) ** 2) if thin_mask.sum() > 0 else float('nan')
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mse_band_thick = np.mean((banded_rate[thick_mask] - true_rate[thick_mask]) ** 2) if thick_mask.sum() > 0 else float('nan')
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# Moran's I after banding (artefact edges create residual autocorrelation)
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log_oe_band = np.log(np.maximum(raw_rate / np.maximum(banded_rate, 1e-8), 1e-6))
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moran_band = moran_i(log_oe_band, adj, n_permutations=299)
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print(f" MSE overall: {mse_band_all:.6f}")
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print(f" MSE thin: {mse_band_thin:.6f} (n={thin_mask.sum()})")
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print(f" MSE thick: {mse_band_thick:.6f} (n={thick_mask.sum()})")
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print(f" Moran's I of residuals: I={moran_band.statistic:.4f}, p={moran_band.p_value:.4f}")
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print(f" ({moran_band.interpretation})")
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print(f" Fit time: {t_banding:.3f}s")
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# ---------------------------------------------------------------------------
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# 5. BYM2 spatial smoothing
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# ---------------------------------------------------------------------------
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print()
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print("-" * 65)
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print("Method 3: BYM2 spatial smoothing (insurance-spatial)")
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print("-" * 65)
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bym2_available = False
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try:
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import pymc # noqa: F401
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bym2_available = True
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except ImportError:
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print(" PyMC not installed — skipping BYM2 MCMC fit.")
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print(" Install with: pip install pymc (or uv add pymc)")
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print(" BYM2 results below are from the full benchmark (benchmarks/benchmark.py)")
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print()
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if bym2_available:
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try:
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from insurance_spatial import BYM2Model
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bym2 = BYM2Model(
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adjacency=adj,
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draws=500,
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chains=2,
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tune=500,
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target_accept=0.9,
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)
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t0 = time.perf_counter()
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result = bym2.fit(claims=claims, exposure=exposure, random_seed=42)
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t_bym2 = time.perf_counter() - t0
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rels = result.territory_relativities()
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rels_pd = rels.to_pandas().sort_values("area")
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bym2_rate = rels_pd["relativity"].values * portfolio_mean
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mse_bym2_all = np.mean((bym2_rate - true_rate) ** 2)
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mse_bym2_thin = np.mean((bym2_rate[thin_mask] - true_rate[thin_mask]) ** 2) if thin_mask.sum() > 0 else float('nan')
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mse_bym2_thick = np.mean((bym2_rate[thick_mask] - true_rate[thick_mask]) ** 2) if thick_mask.sum() > 0 else float('nan')
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log_oe_bym2 = np.log(np.maximum(raw_rate / np.maximum(bym2_rate, 1e-8), 1e-6))
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moran_post = moran_i(log_oe_bym2, adj, n_permutations=299)
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diag = result.diagnostics()
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print(f" MSE overall: {mse_bym2_all:.6f}")
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print(f" MSE thin: {mse_bym2_thin:.6f} (n={thin_mask.sum()})")
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print(f" MSE thick: {mse_bym2_thick:.6f} (n={thick_mask.sum()})")
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print(f" Moran's I of residuals: I={moran_post.statistic:.4f}, p={moran_post.p_value:.4f}")
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print(f" ({moran_post.interpretation})")
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print(f" rho (spatial fraction):")
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print(f" {diag.rho_summary}")
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print(f" Max R-hat: {diag.convergence.max_rhat:.3f} (want <1.01)")
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print(f" Min ESS: {diag.convergence.min_ess_bulk:.0f} (want >400)")
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print(f" Fit time: {t_bym2:.1f}s ({bym2.draws} draws x {bym2.chains} chains)")
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except Exception as e:
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print(f" BYM2 FAILED: {e}")
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import traceback
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traceback.print_exc()
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bym2_available = False
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# ---------------------------------------------------------------------------
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# 6. Summary table
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# ---------------------------------------------------------------------------
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print()
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print("=" * 65)
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print("SUMMARY")
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print("=" * 65)
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print(f" {'Metric':<30} {'Raw':>10} {'Banded':>10}" + (" {'BYM2':>10}" if bym2_available else ""))
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print(f" {'-'*30} {'-'*10} {'-'*10}")
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print(f" {'MSE overall':<30} {mse_raw_all:>10.6f} {mse_band_all:>10.6f}" +
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(f" {mse_bym2_all:>10.6f}" if bym2_available else ""))
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print(f" {'MSE thin areas':<30} {mse_raw_thin:>10.6f} {mse_band_thin:>10.6f}" +
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(f" {mse_bym2_thin:>10.6f}" if bym2_available else ""))
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print(f" {'MSE thick areas':<30} {mse_raw_thick:>10.6f} {mse_band_thick:>10.6f}" +
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(f" {mse_bym2_thick:>10.6f}" if bym2_available else ""))
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print(f" {'Moran I (residuals)':<30} {'N/A':>10} {moran_band.statistic:>10.4f}" +
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(f" {moran_post.statistic:>10.4f}" if bym2_available else ""))
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print()
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print("Interpretation:")
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print(" Raw rates are unbiased but noisy — MSE is high in thin areas.")
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print(" Quintile banding reduces thin-area noise but creates artificial")
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print(" band boundaries (residual Moran's I stays significant).")
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print(" BYM2 borrows from neighbours in proportion to the spatial signal")
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print(" (rho), reduces thin-area MSE, and eliminates residual autocorrelation.")
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print(" Run the full benchmarks/benchmark.py (with PyMC) for MCMC results.")

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