Kolmogorov conditionalization is a strategy for updating credences based on propositions that have initial probability 0. I explore the connection between Kolmogorov conditionalization and Dutch books. Previous discussions of the connection rely crucially upon a factivity assumption: they assume that the agent updates credences based on true propositions. The factivity assumption discounts cases of misplaced certainty, i.e., cases where the agent invests credence 1 in a falsehood. Yet misplaced certainty arises routinely in scientific and philosophical applications of Bayesian decision theory. I prove a non-factive Dutch book theorem and converse Dutch book theorem for Kolmogorov conditionalization. The theorems do not rely upon the factivity assumption, so they establish that Kolmogorov conditionalization has unique pragmatic virtues that persist even in cases of misplaced certainty.

This article discusses how to update one’s credences based on evidence that has initial probability 0. I advance a diachronic norm, Kolmogorov Conditionalization, that governs credal reallocation in many such learning scenarios. The norm is based upon Kolmogorov’s theory of conditional probability. I prove a Dutch book theorem and converse Dutch book theorem for Kolmogorov Conditionalization. The two theorems establish Kolmogorov Conditionalization as the unique credal reallocation rule that avoids a sure loss in the relevant learning scenarios.

The notion of conditional comonotonicity was first used implicitly by Kaas, Dhaene, and Goovaerts (2000) and was formally introduced by Jouini and Napp (2004) as a generalization of the classical concept of comonotonicity. The objective of the present paper is to further investigate this relatively new concept. The main result is that a random vector is comonotonic conditional to a certain σ-field if and only if it is almost surely comonotonic locally on each atom of the conditioning σ-field. We also provide a new proof of a distributional representation and an almost sure representation of a conditionally comonotonic random vector.