Prior Elicitation for Generalised Linear Models and Extensions

TL;DR

This paper introduces a comprehensive Bayesian prior elicitation method for generalized linear models, enabling expert-driven prior specification for complex models with dependencies, extended to various data types, and implemented in R.

Contribution

It presents a novel elicitation approach for priors in GLMs, including dependence modeling with vine copulas, applicable to diverse data types and extended models.

Findings

The method effectively incorporates expert knowledge into priors.

The vine copula dependence model captures substantial prior information.

The approach is demonstrated with a case study on overdispersed count data.

Abstract

A statistical method for the elicitation of priors in Bayesian generalised linear models (GLMs) and extensions is proposed. Probabilistic predictions are elicited from the expert to parametrise a multivariate t prior distribution for the unknown linear coefficients of the GLM and an inverse gamma prior for the dispersion parameter, if unknown. The elicited predictions condition on defined elicitation scenarios. Dependencies among scenarios are then elicited from the expert by additionally conditioning on hypothetical experiments. Elicited conditional medians efficiently parametrise a canonical vine copula model of dependence that may be truncated for efficiency. The statistical elicitation method permits prior parametrisation of GLMs with alternative choices of design matrices or observation models from the same elicitation session. Extensions of the method apply to multivariate data, data with bounded support, semi-continuous data with point mass at zero, and count data with overdispersion or zero-inflation. A case study elicits a prior for an extended GLM embedded in a statistical model of overdispersed counts described by a binomial-simplex mixture distribution. The elicited canonical vine model of dependence is found to incorporate substantial information into the prior. The procedures of the statistical elicitation method are implemented in the R package eglm.