Journal of Applied Probability

Twenty questions with noise: Bayes optimal policies for entropy loss

Bruno Jedynak, Peter I. Frazier, and Raphael Sznitman

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Abstract

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.

Article information

Source
J. Appl. Probab. Volume 49, Number 1 (2012), 114-136.

Dates
First available in Project Euclid: 8 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.jap/1331216837

Digital Object Identifier
doi:10.1239/jap/1331216837

Mathematical Reviews number (MathSciNet)
MR2952885

Zentralblatt MATH identifier
1318.62017

Subjects
Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures 90B40: Search theory 90C39: Dynamic programming [See also 49L20]

Keywords
Twenty questions dynamic programing bisection search object detection entropy loss sequential experimental design Bayesian experimental design

Citation

Jedynak, Bruno; Frazier, Peter I.; Sznitman, Raphael. Twenty questions with noise: Bayes optimal policies for entropy loss. J. Appl. Probab. 49 (2012), no. 1, 114--136. doi:10.1239/jap/1331216837. https://projecteuclid.org/euclid.jap/1331216837.


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