Recovering Posterior Beliefs in Credit Risk: A Latent-State EM Extension of the Information-Geometric Framework

This paper develops a latent-state framework for recovering borrower-level posterior beliefs in credit-risk analysis. Creditworthiness and financial fragility are represented as latent dimensions, while observed borrower scores follow a finite Gaussian mixture model and default depends on the latent profile. Borrower-specific probabilities of default are obtained by averaging profile-specific default probabilities over the posterior distribution of latent states. A joint Expectation--Maximization procedure is used to estimate the mixture structure and the profile-specific default probabilities from observed score--default pairs. After estimation, predictive posterior beliefs are computed using the observed scores alone, thereby preserving the information available before default realization. A controlled simulation experiment evaluates the recovery of structural parameters, posterior beliefs, and borrower-level probabilities of default. Posterior distributions are interpreted as points on the probability simplex, and their recovery is assessed using both conventional error measures and information-geometric divergences. The results provide a controlled benchmark for studying the interaction between latent economic structure, posterior uncertainty, and credit-risk prediction.