Prior sample size extensions for assessing prior impact and prior--likelihood discordance

TL;DR

This paper develops a comprehensive framework for quantifying the influence of prior distributions on Bayesian inference, extending the concept of prior sample size to handle non-conjugate, conflicting, and negative prior impacts, with practical and theoretical insights.

Contribution

It introduces new methods to estimate and interpret prior sample size beyond conjugate models, including a relative measure as a function of data size and the allowance of negative values to indicate conflict.

Findings

Asymptotic formula for prior impact involving centrality and spread measures

Graphical diagnosis of prior-likelihood discordance

Application to real data on lupus nephritis

Abstract

This paper outlines a framework for quantifying the prior's contribution to posterior inference in the presence of prior-likelihood discordance, a broader concept than the usual notion of prior-likelihood conflict. We achieve this dual purpose by extending the classic notion of \textit{prior sample size}, $M$, in three directions: (I) estimating $M$ beyond conjugate families; (II) formulating $M$ as a relative notion, i.e., as a function of the likelihood sample size $k, M(k),$ which also leads naturally to a graphical diagnosis; and (III) permitting negative $M$, as a measure of prior-likelihood conflict, i.e., harmful discordance. Our asymptotic regime permits the prior sample size to grow with the likelihood data size, hence making asymptotic arguments meaningful for investigating the impact of the prior relative to that of likelihood. It leads to a simple asymptotic formula for quantifying the impact of a proper prior that only involves computing a centrality and a spread measure of the prior and the posterior. We use simulated and real data to illustrate the potential of the proposed framework, including quantifying how weak is a "weakly informative" prior adopted in a study of lupus nephritis. Whereas we take a pragmatic perspective in assessing the impact of a prior on a given inference problem under a specific evaluative metric, we also touch upon conceptual and theoretical issues such as using improper priors and permitting priors with asymptotically non-vanishing influence.