Declarations of independence

Synthese 194 (10):3979-3995 (2017)

Abstract

According to orthodox (Kolmogorovian) probability theory, conditional probabilities are by definition certain ratios of unconditional probabilities. As a result, orthodox conditional probabilities are undefined whenever their antecedents have zero unconditional probability. This has important ramifications for the notion of probabilistic independence. Traditionally, independence is defined in terms of unconditional probabilities (the factorization of the relevant joint unconditional probabilities). Various “equivalent” formulations of independence can be given using conditional probabilities. But these “equivalences” break down if conditional probabilities are permitted to have conditions with zero unconditional probability. We reconsider probabilistic independence in this more general setting. We argue that a less orthodox but more general (Popperian) theory of conditional probability should be used, and that much of the conventional wisdom about probabilistic independence needs to be rethought.

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Branden Fitelson
Northeastern University

References found in this work

The Logic of Scientific Discovery.Karl Raimund Popper - 1934 - London, England: Routledge.
Logical Foundations of Probability.Rudolf Carnap - 1950 - Chicago, IL, USA: Chicago University of Chicago Press.
The Logic of Scientific Discovery.Karl Popper - 1959 - Studia Logica 9:262-265.
A Subjectivist’s Guide to Objective Chance.David K. Lewis - 1980 - In Richard C. Jeffrey (ed.), Studies in Inductive Logic and Probability, Volume II. Berkeley: University of California Press. pp. 263-293.

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Citations of this work

Conditional Probabilities.Kenny Easwaran - 2019 - In Richard Pettigrew & Jonathan Weisberg (eds.), The Open Handbook of Formal Epistemology. PhilPapers Foundation. pp. 131-198.
Conditional Degree of Belief and Bayesian Inference.Jan Sprenger - 2020 - Philosophy of Science 87 (2):319-335.
You’Ve Come a Long Way, Bayesians.Jonathan Weisberg - 2015 - Journal of Philosophical Logic 44 (6):817-834.

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