The Wasserstein Impact Measure (WIM): a generally applicable, practical tool for quantifying prior impact in Bayesian statistics

TL;DR

This paper introduces the Wasserstein Impact Measure (WIM), a practical and versatile tool for quantifying prior impact in Bayesian analysis, overcoming previous limitations of the theoretical Wasserstein approach.

Contribution

The paper develops WIM, a practical version of the Wasserstein impact measure that works without nested priors, complex posteriors, or scalar restrictions, and demonstrates its effectiveness.

Findings

WIM closely matches the theoretical Wasserstein impact in simulations

WIM outperforms existing prior impact measures in various scenarios

WIM is versatile and applicable to real datasets

Abstract

The prior distribution is a crucial building block in Bayesian analysis, and its choice will impact the subsequent inference. It is therefore important to have a convenient way to quantify this impact, as such a measure of prior impact will help us to choose between two or more priors in a given situation. A recently proposed approach consists in determining the Wasserstein distance between posteriors resulting from two distinct priors, revealing how close or distant they are. In particular, if one prior is the uniform/flat prior, this distance leads to a genuine measure of prior impact for the other prior. While highly appealing and successful from a theoretical viewpoint, this proposal suffers from practical limitations: it requires prior distributions to be nested, posterior distributions should not be of a too complex form, in most considered settings the exact distance was not computed but sharp upper and lower bounds were proposed, and the proposal so far is restricted to scalar parameter settings. In this paper, we overcome all these limitations by introducing a practical version of this theoretical approach, namely the Wasserstein Impact Measure (WIM). In three simulated scenarios, we will compare the WIM to the theoretical Wasserstein approach, as well as to two competitor prior impact measures from the literature. We finally illustrate the versatility of the WIM by applying it on two datasets.