Ideal-Theoretic Strategies for Asymptotic Approximation of Marginal Likelihood Integrals

TL;DR

This paper introduces algebraic geometry methods, specifically real log canonical thresholds, to asymptotically evaluate marginal likelihood integrals in Bayesian models, including finite state models.

Contribution

It develops effective algebraic geometry techniques for computing asymptotic approximations of marginal likelihoods, extending Watanabe's approach.

Findings

Effective methods for computing real log canonical thresholds.

Application to finite state discrete models.

Resolution of singularities enhances approximation accuracy.

Abstract

The accurate asymptotic evaluation of marginal likelihood integrals is a fundamental problem in Bayesian statistics. Following the approach introduced by Watanabe, we translate this into a problem of computational algebraic geometry, namely, to determine the real log canonical threshold of a polynomial ideal, and we present effective methods for solving this problem. Our results are based on resolution of singularities. They apply to parametric models where the Kullback-Leibler distance is upper and lower bounded by scalar multiples of some sum of squared real analytic functions. Such models include finite state discrete models.