A Dirichlet Mixed-Membership Model for Exact Multivariate Distributional Credibility

Paper: arXiv 2610.05741

Authors: Sebastián Calcetero Vanegas, Ian Weng Chan

Abstract

Credibility theory combines individual experience with portfolio information for insurance pricing, but classical formulations focus primarily on conditional means and expected premiums. We propose a Dirichlet mixed-membership model (DMMM) for multivariate distributional credibility. Policyholder risk is represented by a stable composition over latent risk classes: baseline characteristics determine its a priori assessment, while repeated multivariate experience progressively updates its a posteriori assessment. Support-specific expert distributions accommodate heterogeneous outcomes, with dependence induced through the shared latent structure. The resulting posterior predictive distribution can be written exactly as a convex combination of portfolio and experience components, extending the familiar credibility-factor structure beyond the conditional mean. We study the framework through a simulation experiment and a vehicle telematics application. The simulation shows that relatively simple experts can capture complex multivariate insurance distributions and learn policyholder-specific risk as experience accumulates. In the telematics application, recent claims and driving behaviour provide complementary information for future claim-frequency prediction, allowing similar policyholders to receive different experience-rated assessments. The DMMM therefore provides a flexible and interpretable framework for combining heterogeneous insurance experience while retaining the structure of classical credibility.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper introduces a novel and mathematically sophisticated Dirichlet mixed-membership model for multivariate distributional credibility, extending classical credibility theory. It demonstrates strong empirical rigor through both simulation and a real-world telematics application, providing a flexible and interpretable framework for insurance pricing.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Credibility Theory for Exact Multivariate Distributional Credibility] --> B{Key Methodology: Dirichlet Mixed-Membership Model (DMMM)};
    B --> C[Data/Inputs: Portfolio Information, Individual Policyholder Experience (Multivariate), Baseline Characteristics, Support-Specific Expert Distributions];
    C --> D{Computational Process: Bayesian Inference for DMMM, Posterior Predictive Distribution Calculation};
    D --> E[Outcomes: Exact Convex Combination of Portfolio & Experience Components (Credibility Factor Structure)];
    E --> F[Key Findings: Model Captures Complex Multivariate Distributions, Learns Policyholder-Specific Risk (Simulation), Complementary Information from Telematics for Claim Prediction, Flexible & Interpretable Framework];

Cited References (39)

1. Experience Rating and Credibility (1967) — link
2. A Course in Credibility Theory and Its Applications (2005)
3. Glaubw "u rdigkeit f "u r Schadens "a tze (1970)
4. Severity Estimation in Dependent Collective Risk Models (2026) — link
5. Credibility Estimation of Distribution Functions with Applications to Experience Rating in General Insurance (2015) — link
6. Effective Experience Rating for Large Insurance Portfolios via Surrogate Modeling (2024) — link
7. Stan: A Probabilistic Programming Language (2017) — link
8. Bayesian Credibility under a Bivariate Prior on the Frequency and the Severity of Claims (2021) — link

0 of 39 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.