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Conditioning using conditional expectations: The Borel-Kolmogorov Paradox

Gyenis, Zalán and Rédei, Miklós and Hofer-Szabó, Gábor (2015) Conditioning using conditional expectations: The Borel-Kolmogorov Paradox. [Preprint]

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Abstract

The Borel-Kolmogorov Paradox is typically taken to highlight a tension between our intuition that certain conditional probabilities with respect to probability zero conditioning events are well defined and the mathematical definition of conditional probability by Bayes' formula, which looses its meaning when the conditioning event has probability zero. We argue in this paper that the theory of conditional expectations is the proper mathematical device to conditionalize, and this theory allows conditionalization with respect to probability zero events. The conditional probabilities on probability zero events in the Borel-Kolmogorov Paradox also can be calculated using conditional expectations. The alleged clash arising from the fact that one obtains different values for the conditional probabilities on probability zero events depending on what conditional expectation one uses to calculate them is resolved by showing that the different conditional probabilities obtained using different conditional expectations cannot be interpreted as calculating in different parametrizations of the conditional probabilities of the same event with respect to the same conditioning conditions. We conclude that there is no clash between the correct intuition about what the conditional probabilities with respect to probability zero events are and the technically proper concept of conditionalization via conditional expectations -- the Borel-Kolmogorov Paradox is just a pseudo-paradox.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
Gyenis, Zalángyz@renyi.hu
Rédei, MiklósM.Redei@lse.ac.uk
Hofer-Szabó, Gáborgsz@szig.hu
Additional Information: Submitted for publication
Keywords: Conditionalization, Borel-Kolmogorov Paradox, Interpretation of probability
Subjects: Specific Sciences > Probability/Statistics
Depositing User: Dr Miklos Redei
Date Deposited: 15 Mar 2015 17:13
Last Modified: 15 Mar 2015 17:13
Item ID: 11377
Subjects: Specific Sciences > Probability/Statistics
Date: 13 March 2015
URI: http://philsci-archive.pitt.edu/id/eprint/11377

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