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Disintegration and Bayesian inversion via string diagrams

Published online by Cambridge University Press:  13 March 2019

Kenta Cho Open the ORCID record for Kenta Cho [Opens in a new window]  and
Bart Jacobs
Kenta Cho*
Affiliation:
Information Systems Architecture Science Research Division, National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
Bart Jacobs
Affiliation:
Digital Security Group, Institute for Computing and Information Sciences, Radboud University, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands
*
*Corresponding author. Email: cho@nii.ac.jp
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Abstract

The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability – via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.

Keywords

Conditional probabilitydisintegrationBayesian inversionstring diagrammonoidal category

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Issue 7 , August 2019 , pp. 938 - 971
Copyright
© Cambridge University Press 2019 

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