Bayesian defeat of certainties

Abstract

When P(E) > 0, conditional probabilities P(H|E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(H|E)$$\\end{document} are given by the ratio formula. An agent engages in ratio conditionalization when she updates her credences using conditional probabilities dictated by the ratio formula. Ratio conditionalization cannot eradicate certainties, including certainties gained through prior exercises of ratio conditionalization. An agent who updates her credences only through ratio conditionalization risks permanent certainty in propositions against which she has overwhelming evidence. To avoid this undesirable consequence, I argue that we should supplement ratio conditionalization with Kolmogorov conditionalization, a strategy for updating credences based on propositions E such that P(E) = 0. Kolmogorov conditionalization can eradicate certainties, including certainties gained through prior exercises of conditionalization. Adducing general theorems and detailed examples, I show that Kolmogorov conditionalization helps us model epistemic defeat across a wide range of circumstances.