Fast Measurements of Robustness to Changing Priors in Variational Bayes

TL;DR

This paper introduces efficient methods to quantify the robustness of Bayesian posteriors to prior changes using Variational Bayes, including local and non-local measures, demonstrated on simulated data.

Contribution

It provides the first practical, easy-to-compute robustness measures for VB posteriors, including a closed-form influence function and an approximate non-local measure.

Findings

Closed-form influence function for VB posteriors

Effective approximate non-local robustness measure

Demonstrated on simulated data

Abstract

In Bayesian analysis, the posterior follows from the data and a choice of a prior and a likelihood. One hopes that the posterior is robust to reasonable variation in the choice of prior, since this choice is made by the modeler and is often somewhat subjective. A different, equally subjectively plausible choice of prior may result in a substantially different posterior, and so different conclusions drawn from the data. Were this to be the case, our conclusions would not be robust to the choice of prior. To determine whether our model is robust, we must quantify how sensitive our posterior is to perturbations of our prior. Despite the importance of the problem and a considerable body of literature, generic, easy-to-use methods to quantify Bayesian robustness are still lacking. Abstract In this paper, we demonstrate that powerful measures of robustness can be easily calculated from Variational Bayes (VB) approximate posteriors. We begin with local robustness, which measures the effect of infinitesimal changes to the prior on a posterior mean of interest. In particular, we show that the influence function of Gustafson (2012) has a simple, easy-to-calculate closed form expression for VB approximations. We then demonstrate how local robustness measures can be inadequate for non-local prior changes, such as replacing one prior entirely with another. We propose a simple approximate non-local robustness measure and demonstrate its effectiveness on a simulated data set.