Dirichlet Mixed-Membership Model for Exact Multivariate Distributional Credibility

01 ABSTRACT

Credibility theory combines individual experience with portfolio information for insurance pricing, but classical formulations focus primarily on conditional means and expected premiums. The authors propose a Dirichlet mixed-membership model (DMMM) for multivariate distributional credibility, representing policyholder risk as a stable composition over latent risk classes, with baseline characteristics determining the a priori assessment and repeated multivariate experience progressively updating the a posteriori assessment. The resulting posterior predictive distribution can be written exactly as a convex combination of portfolio and experience components, extending the credibility-factor structure beyond the conditional mean.

02 KEY FINDINGS

1. DMMM represents policyholder risk as a stable composition over latent risk classes via a Dirichlet mixed-membership structure, with baseline characteristics determining the prior and multivariate experience updating the posterior.
2. Support-specific expert distributions accommodate heterogeneous outcomes, dependence is induced through the shared latent structure, and the posterior predictive distribution is exactly a convex combination of portfolio and experience components.
3. Simulation shows that relatively simple experts can capture complex multivariate insurance distributions and learn policyholder-specific risk as experience accumulates; in a vehicle telematics application, recent claims and driving behaviour provide complementary information.