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KOLMOGOROV CONDITIONALIZERS CAN BE DUTCH BOOKED (IF AND ONLY IF THEY ARE EVIDENTIALLY UNCERTAIN)

Part of: Mathematical economics Decision theory Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  21 October 2020

ALEXANDER MEEHAN  and
SNOW ZHANG
ALEXANDER MEEHAN
Affiliation:
DEPARTMENT OF PHILOSOPHY PRINCETON UNIVERSITY 1879 HALL PRINCETON, NJ08544, USAE-mail: alexandermeehan@princeton.eduE-mail: xueyinz@princeton.edu
SNOW ZHANG
Affiliation:
DEPARTMENT OF PHILOSOPHY PRINCETON UNIVERSITY 1879 HALL PRINCETON, NJ08544, USAE-mail: alexandermeehan@princeton.eduE-mail: xueyinz@princeton.edu
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Abstract

A vexing question in Bayesian epistemology is how an agent should update on evidence which she assigned zero prior credence. Some theorists have suggested that, in such cases, the agent should update by Kolmogorov conditionalization, a norm based on Kolmogorov’s theory of regular conditional distributions. However, it turns out that in some situations, a Kolmogorov conditionalizer will plan to always assign a posterior credence of zero to the evidence she learns. Intuitively, such a plan is irrational and easily Dutch bookable. In this paper, we propose a revised norm, Kolmogorov–Blackwell conditionalization, which avoids this problem. We prove a Dutch book theorem and converse Dutch book theorem for this revised norm, and relate our results to those of Rescorla (2018).

Keywords

conditionalizationDutch booksformal epistemologyregular conditional distributionsevidential certainty

MSC classification

Primary: 03A99: None of the above, but in this section
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures 91B06: Decision theory
Type
Research Article
Information
The Review of Symbolic Logic , Volume 15 , Issue 3 , September 2022 , pp. 722 - 757
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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