The information loss of a stochastic map
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We provide a stochastic extension of the Baez-Fritz-Leinster characterization of the Shannon information loss associated with a measure-preserving function. This recovers the conditional entropy and a closely related information-theoretic measure that we call `conditional information loss.' Although not functorial, these information measures are semi-functorial, a concept we introduce that is definable in any Markov category. We also introduce the notion of an `entropic Bayes' rule' for information measures, and we provide a characterization of conditional entropy in terms of this rule.
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