Exact Posterior Mean and Covariance for Generalized Linear Mixed Models

TL;DR

This paper introduces the SIC method to compute exact posterior mean and covariance of random effects in GLMMs without relying on intractable integrals or Monte Carlo simulations, advancing Bayesian analysis.

Contribution

The paper presents a novel optimization-based SIC method that computes exact posterior moments in GLMMs with non-normal responses, bypassing intractable integrals.

Findings

SIC method provides exact posterior mean and covariance.

SIC avoids the need for posterior distribution computation.

Method applicable to a wide class of GLMMs with non-normal responses.

Abstract

A novel method is proposed for the exact posterior mean and covariance of the random effects given the response in a generalized linear mixed model (GLMM) when the response does not follow normal. The research solves a long-standing problem in Bayesian statistics when an intractable integral appears in the posterior distribution. It is well-known that the posterior distribution of the random effects given the response in a GLMM when the response does not follow normal contains intractable integrals. Previous methods rely on Monte Carlo simulations for the posterior distributions. They do not provide the exact posterior mean and covariance of the random effects given the response. The special integral computation (SIC) method is proposed to overcome the difficulty. The SIC method does not use the posterior distribution in the computation. It devises an optimization problem to reach the task. An advantage is that the computation of the posterior distribution is unnecessary. The proposed SIC avoids the main difficulty in Bayesian analysis when intractable integrals appear in the posterior distribution.