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Twenty Questions with Noise: Bayes Optimal Policies for Entropy Loss

Part of: Operations research and management science Decision theory Markov processes

Published online by Cambridge University Press:  04 February 2016

Bruno Jedynak ,
Peter I. Frazier  and
Raphael Sznitman
Bruno Jedynak*
Affiliation:
Johns Hopkins University
Peter I. Frazier*
Affiliation:
Cornell University
Raphael Sznitman*
Affiliation:
Johns Hopkins University
*
∗ Postal address: Department of Applied Mathematics and Statistics, Johns Hopkins University, Whitehead 208-B, 3400 North Charles Street, Baltimore, MD 21218, USA. Email address: bruno.jedynak@jhu.edu
∗∗ Postal address: School of Operations Research and Industrial Engineering, Cornell University, 232 Rhodes Hall, Ithaca, NY 14850, USA.
∗∗∗ Postal address: Johns Hopkins University, Hackerman Hall, 3400 North Charles Street, Baltimore, MD 21218, USA.
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Abstract

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We consider the problem of twenty questions with noisy answers, in which we seek to find a target by repeatedly choosing a set, asking an oracle whether the target lies in this set, and obtaining an answer corrupted by noise. Starting with a prior distribution on the target's location, we seek to minimize the expected entropy of the posterior distribution. We formulate this problem as a dynamic program and show that any policy optimizing the one-step expected reduction in entropy is also optimal over the full horizon. Two such Bayes optimal policies are presented: one generalizes the probabilistic bisection policy due to Horstein and the other asks a deterministic set of questions. We study the structural properties of the latter, and illustrate its use in a computer vision application.

Keywords

Twenty questionsdynamic programingbisectionsearchobject detectionentropy losssequential experimental designBayesian experimental design

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures 90B40: Search theory 90C39: Dynamic programming
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 1 , March 2012 , pp. 114 - 136
Copyright
© Applied Probability Trust 

Footnotes

Research supported in part by AFOSRYIP FA9550-11-1-0083.

Research supported in part by NIH grant R01 EB 007969-01.

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