ParaEval and the CRP/HDP Model: Bayesian Nonparametric Trigger Calibration for Parametric Insurance

A parametric insurance policy pays when an observable index — wind speed, rainfall accumulation, flood depth — crosses a contractual threshold. The entire product hinges on that threshold being correctly calibrated: set it too low and the insurer overpays (false positives); set it too high and the policyholder suffers uncompensated loss (false negatives). The sum of these two errors is basis risk, and it is the single most important quality metric for any parametric product. Industry practice typically calibrates thresholds using Generalized Extreme Value (GEV) distributions, Weibull fits, or historical quantiles. These methods assume a single, stationary generating process — an assumption that breaks when peril events cluster into distinct regimes (e.g., fast-track high-wind typhoons vs. slow-moving rain-dominant systems) and when climate non-stationarity shifts those regimes over time. ParaEval is a decision-evaluation platform built to surface exactly this kind of structural mismatch. Its CRP/HDP sub-model replaces the single-distribution assumption with a Bayesian nonparametric mixture that discovers latent peril regimes directly from data, then derives trigger thresholds from the boundaries between loss-producing and non-loss-producing regimes.

ParaEval is not a model — it is a decision-evaluation platform that sits between raw event data and the payout recommendation. It structures the entire evidence-to-decision pipeline into four composable stages: evidence ingestion, trigger evaluation, settlement logic, and audit trail generation. Each stage is deterministic given its inputs — the same evidence snapshot always produces the same decision, reasoning trace, and payout recommendation. This determinism is not a convenience; it is a regulatory requirement. Parametric insurance settlements must be reproducible and explainable to auditors, regulators, and dispute panels. The system handles multiple evidence types (weather API readings, satellite observations, uploaded documents, claims adjuster reports) and classifies each source by reliability tier: authoritative (the contractual index source), corroborating (independent sources that directionally agree), and indicative (weaker signals useful for basis-risk analysis). The decision engine applies four rule checks — authoritative source coverage, contract threshold test, cross-source corroboration, and counter-signal management — before producing a confidence score, trigger status (met / borderline / not_met), and a structured settlement memo.

The Chinese Restaurant Process (CRP) is a constructive definition of the Dirichlet Process (DP) that makes its clustering behavior intuitive. Imagine a restaurant with infinitely many tables. The first customer sits at table 1. Each subsequent customer either joins an existing table with probability proportional to the number of people already seated there, or starts a new table with probability proportional to a concentration parameter alpha. This process generates a random partition of customers into groups — and it does so without fixing the number of groups in advance. In the context of peril modeling, each "customer" is a historical typhoon event and each "table" is a latent peril regime. The CRP prior encodes a rich-get-richer dynamic: large regimes attract more events, but the concentration parameter alpha controls how readily new regimes are created. Crucially, alpha is not fixed — we place a Gamma prior on it and infer its posterior value alongside the regime assignments. A higher posterior alpha means the data is better explained by more regimes; a lower alpha means fewer, larger clusters suffice. This is the Escobar & West (1995) auxiliary variable method, and it gives the model an automatic Occam's razor: it discovers exactly as many regimes as the data warrants.

We implement collapsed Gibbs sampling following Neal (2000, Algorithm 3). "Collapsed" means we analytically integrate out the cluster-specific parameters ($\mu_k$, $\Sigma_k$) using the Normal-Inverse-Wishart conjugacy, and sample only the discrete assignment variables $z$ and the concentration parameter $\alpha$. This reduces the state space dramatically — instead of sampling $O(K \cdot d^2)$ continuous parameters per iteration, we sample $O(n)$ discrete assignments. The collapsed sampler mixes faster and converges in fewer iterations. Each Gibbs iteration makes a full pass over all $n$ events. For event $i$, we temporarily remove it from its current cluster, compute the CRP conditional probability of assigning it to each existing cluster (using the Student-$t$ predictive distribution that falls out of the NIW posterior) and to a new cluster (using the prior predictive), then sample from the resulting categorical distribution. After the full pass, we update $\alpha$ via the Escobar & West auxiliary variable method. Convergence is monitored via the split-chain $\hat{R}$ diagnostic on $K$ (the number of active clusters) and the log-likelihood trace.

A typhoon is not a single peril — it produces wind damage, rainfall flooding, and storm surge simultaneously. The conditional distribution of secondary peril intensity (e.g., flood depth) given the primary peril context (e.g., Saffir-Simpson category) is critical for multi-trigger products. The Hierarchical Dirichlet Process (HDP) extends the CRP to grouped data: each typhoon category gets its own Dirichlet Process mixture for flood depth, but all category-specific mixtures share a common set of global atoms drawn from a top-level DP. This sharing mechanism is the key statistical insight — it allows rare categories (Cat 5 events) to borrow strength from more common categories (Cat 2-3 events) through shared mixture components, while still allowing category-specific distributional differences. In the ParaEval implementation, this is approximated using context-specific Gaussian Mixture Models with a global backoff model for sparse categories, following the MAP-sharing approximation rather than a full Gibbs HDP sampler (Teh et al., 2006). This is a tractable compromise for the dataset sizes typical in APAC typhoon catalogs (~500-2000 events).

The CRP sampler discovers latent regimes. The trigger calibrator turns those regimes into actionable insurance thresholds. The strategy is conceptually simple: for each posterior sample of regime assignments, identify which regimes are loss-producing (majority of events have loss_occurred = True) and which are not. For each peril feature, find the 1D decision boundary between the closest loss-regime and no-loss-regime pair. This boundary is the trigger threshold theta* for that feature in that posterior sample. Repeating across all posterior samples yields a distribution of theta* values, from which we extract the posterior mean as the point estimate and the 5th/95th percentiles as a 90% credible interval. This credible interval is the key output that traditional methods cannot produce: it directly quantifies the uncertainty in the trigger threshold due to finite data and regime assignment uncertainty. A wide CI on theta* signals that the trigger boundary is poorly resolved — a direct warning to the product designer that basis risk may be high.

Traditional catastrophe models assume stationarity — the same generating process that produced historical events will produce future ones. Climate change violates this assumption. The alpha-drift index is a novel application of the CRP concentration parameter as a non-stationarity detector. The idea is straightforward: fit the CRP sampler on rolling time windows (e.g., 5-year windows sliding across a 50-year catalog) and track the posterior mean of alpha over time. A rising alpha indicates that events in recent windows are increasingly poorly explained by the regime structure of earlier windows — the model needs more clusters to fit the data, which means new peril patterns are emerging. A falling alpha means the regime structure is consolidating. We apply the Mann-Kendall trend test to the alpha time series. A statistically significant positive trend (p < 0.05) is a formal signal that the historical calibration basis is becoming stale — trigger thresholds should be recalibrated on more recent data, or the product should be repriced to account for structural uncertainty. The alpha-drift index can also be correlated with external climate indices (e.g., sea surface temperature anomalies, ENSO phase) to test hypotheses about the physical drivers of regime shift.

We evaluate the CRP/HDP trigger against four standard actuarial methods on the same synthetic event corpus (500 events, 3 embedded true regimes, 80/20 train-test split). All methods are evaluated on identical test events using the same loss labels. The primary metric is basis risk (FP + FN)/N — lower is better. We also report precision (fraction of trigger activations that correspond to actual losses), recall (fraction of actual losses captured by the trigger), and boundary F1 (the harmonic mean of precision and recall, directly analogous to the boundary F1 reported in morpheme segmentation research). The CRP/HDP model consistently achieves the lowest basis risk because it calibrates thresholds against cluster boundaries rather than distributional quantiles — it "sees" the regime structure that quantile-based methods average over.

Evaluation on real IBTrACS + ERA5 + EM-DAT data requires CDS API access and EM-DAT registration. For reproducibility, the demo mode uses a synthetic corpus with three embedded regimes that mirror real APAC typhoon archetypes. Regime 0 (35% of events) represents fast-track, high-wind storms (Cat 4-5, mean wind 55 m/s, mean rain 80mm, fast translation at 28 km/h). Regime 1 (40%) represents slow-moving rain-dominant systems (Cat 2-3, mean wind 38 m/s, mean rain 250mm, slow translation at 10 km/h). Regime 2 (25%) represents surge-dominant coastal storms (Cat 3-4, mean wind 45 m/s, mean surge 4.2m, moderate translation at 18 km/h). Loss amounts are regime-correlated: wind-driven losses for Regime 0, flood-driven for Regime 1, surge-driven for Regime 2. The loss_occurred label is set at the median economic loss, giving a balanced 50% trigger rate — a worst-case scenario for basis risk (hardest to separate). Six peril features span the multivariate event space: maximum sustained wind (m/s), 24h rainfall (mm), storm surge (m), minimum central pressure (hPa), translation speed (km/h), and track deviation from climatological mean (km).

The CRP/HDP model plugs into ParaEval through a model adapter layer. When an analyst runs CRP analysis on a case, the system executes a full calibration pipeline (simulation + threshold calibration + evaluation) and produces a structured evidence item that is injected into the case's evidence stack. This evidence item carries the calibrated threshold (theta*), the model's recommendation, and an explicit reliability tag of "indicative" — meaning the CRP output informs the decision but does not override authoritative index readings. The recommendation text is auto-generated based on the calibration metrics: if boundary F1 > 0.7 and basis risk < 0.3, the model output is flagged as "strong trigger discrimination"; if F1 > 0.5, it is "directionally useful"; below that, it is "weak for this feature." This tiered framing prevents over-reliance on model outputs that happen to be poorly calibrated for the specific peril feature and event geography. The analyst sees the CRP output alongside the weather API readings, satellite observations, and adjuster reports — as one signal among several, with its uncertainty clearly surfaced.

The CRP/HDP model draws on foundational work in Bayesian nonparametrics and applies it to a novel domain — parametric insurance trigger calibration. These references cover the mathematical foundations, the inference algorithms, and the insurance context.