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Entropy 2017, 19(8), 402;

Optimal Belief Approximation

Max-Planck-Institut für Astrophysik, Karl-Schwarzschildstr. 1, 85748 Garching, Germany
Ludwig-Maximilians-Universität München, Geschwister-Scholl-Platz 1, 80539 Munich, Germany
Author to whom correspondence should be addressed.
Received: 18 April 2017 / Revised: 4 July 2017 / Accepted: 5 July 2017 / Published: 4 August 2017
(This article belongs to the Section Information Theory, Probability and Statistics)
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In Bayesian statistics probability distributions express beliefs. However, for many problems the beliefs cannot be computed analytically and approximations of beliefs are needed. We seek a loss function that quantifies how “embarrassing” it is to communicate a given approximation. We reproduce and discuss an old proof showing that there is only one ranking under the requirements that (1) the best ranked approximation is the non-approximated belief and (2) that the ranking judges approximations only by their predictions for actual outcomes. The loss function that is obtained in the derivation is equal to the Kullback-Leibler divergence when normalized. This loss function is frequently used in the literature. However, there seems to be confusion about the correct order in which its functional arguments—the approximated and non-approximated beliefs—should be used. The correct order ensures that the recipient of a communication is only deprived of the minimal amount of information. We hope that the elementary derivation settles the apparent confusion. For example when approximating beliefs with Gaussian distributions the optimal approximation is given by moment matching. This is in contrast to many suggested computational schemes. View Full-Text
Keywords: information theory; Bayesian inference; loss function; axiomatic derivation; machine learning information theory; Bayesian inference; loss function; axiomatic derivation; machine learning

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Leike, R.H.; Enßlin, T.A. Optimal Belief Approximation. Entropy 2017, 19, 402.

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