Another Approach to Consensus and Maximally Informed Opinions with Increasing Evidence

Abstract

Merging of opinions results underwrite Bayesian rejoinders to complaints about the subjective nature of personal probability. Such results establish that sufficiently similar priors achieve consensus in the long run when fed the same increasing stream of evidence. Initial subjectivity, the line goes, is of mere transient significance, giving way to intersubjective agreement eventually. Here, we establish a merging result for sets of probability measures that are updated by Jeffrey conditioning. This generalizes a number of different merging results in the literature. We also show that such sets converge to a shared, maximally informed opinion. Convergence to a maximally informed opinion is a (weak) Jeffrey conditioning analogue of Bayesian "convergence to the truth" for conditional probabilities. Finally, we demonstrate the philosophical significance of our study by detailing applications to the topics of dynamic coherence, imprecise probabilities, and probabilistic opinion pooling.

Distention for Sets of Probabilities

“…Schervish and Seidenfeld establish that closed, convex sets of mutually absolutely continuous probabilities that are generated by finitely many extreme points merge under Bayesian conditionalization (Schervish and Seidenfeld, 1990, Corollary 1). In previous work, we generalize this result, showing that closed, convex sets of mutually absolutely continuous probabilities that are generated by finitely many extreme points merge under Jeffrey conditioning as well (Stewart and Nielsen, 2019, Proposition 1). 13 For such sets of distributions, the significance of distention depends on the importance of the short run.…”

A Dilemma for Solomonoff Prediction

“… 14 This and related results are discussed extensively by Earman (1992), Huttegger (2015), and Nielsen and Stewart (2018, 2019). …”

Persistent Disagreement and Polarization in a Bayesian Setting

“…He writes, "experience trumps any initial belief state; diverging opinions are just a sign that not enough evidence has accumulated yet" (2015, p. 613). Savage (1954), Blackwell and Dubins (1962), and Gaifman and Snir (1982) provide classic versions of merging of opinions theorems, which have since been generalized in various ways (Schervish and Seidenfeld, 1990;Huttegger, 2015;Stewart and Nielsen, 2018). Such classic versions of these results attained their prominent theoretical status due to the subjective nature of personal probabilities.…”