Support for Geometric Pooling

Abstract

Supra-Bayesianism is the Bayesian response to learning the opinions of others. Probability pooling constitutes an alternative response. One natural question is whether there are cases where probability pooling gives the supra-Bayesian result. This has been called the problem of Bayes-compatibility for pooling functions. It is known that in a common prior setting, under standard assumptions, linear pooling cannot be non-trivially Bayes-compatible. We show by contrast that geometric pooling can be non-trivially Bayes-compatible. Indeed, we show that, under certain assumptions, geometric and Bayes-compatible pooling are equivalent. Granting supra-Bayesianism its usual normative status, one upshot of our study is thus that, in a certain class of epistemic contexts, geometric pooling enjoys a normative advantage over linear pooling as a social learning mechanism. We discuss the philosophical ramifications of this advantage, which we show to be robust to variations in our statement of the Bayes-compatibility problem.

An Interpretation of Weights in Linear Opinion Pooling

“…Bradley (2018) shows that assuming the linear pool, the common prior and the principle of deference entail that the expert opinions need to be equal, a point further refined and corrected in Dawid and Mortera (2019). Baccelli and Stewart (2020) develop these results and argue that the linear pool is inconsistent with Bayesian models that comply to deference. They go on to prove that under certain additional assumptions the geometric pool is in fact the only coherent pooling method.…”

“…What the current paper achieves for the linear pool, namely a new interpretation of the pooling weights through a Bayesian model of the pooling operation, we can also imagine for other pooling methods. In the Bayesian representation of geometric pooling, following Russell et al (2015), Dietrich and List (2016), EaGlHiVe (2016) 8 and Baccelli and Stewart (2020), the weights show up in the likelihood functions for the revealed opinions of others, directly modifying their impact. A clean expression of the weights in terms of belief can perhaps be distilled from these likelihood functions.…”

“…Other contributions focus on the ways in which linear pooling fails to accommodate the specifics of learning multiple expert opinions (Bradley 2006(Bradley , 2007(Bradley , 2018Steele 2012). In general, Bayesian models cover a wider range of social deliberations than models based on the linear opinion pool, and the latter models violate various natural constraints on social learning (e.g., Dawid et al 1995;Bradley 2018;Dawid and Mortera 2019;Baccelli and Stewart 2020).…”

Geometric Pooling: A User's Guide

“…How should you update your probabilities in those sequences? Baccelli and Stewart ([2023]) argue that geometric pooling is a good strategy in this case. Let's suppose you and your fellow shared the same prior probability function P before you started collecting evidence.…”

“…In this article, we consider a particular proposal: you should combine your probabilities with your fellow's using a method known as geometric pooling. We begin in section 3 by raising several objections to a recent argument in favour of geometric pooling due to Baccelli and Stewart ([2023]). Then we turn, in sections 4 and 5, to what we take to be better arguments in its favour.…”