Which Boolean Functions are Most Informative?

Kumar, Gowtham; Courtade, Thomas A.

doi:10.48550/arxiv.1302.2512

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“…We now present a few direct but powerful corollaries of the result in Theorem 3. We note that a bound similar to (16) below has appeared in the context of bounding the minmax decision risk in [25, (3.4)]. However, the proof technique used in [25] does not lead to the general bound presented in Theorem 3.…”

Section: Remark 2 If λmentioning

confidence: 97%

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Bounds on inference

Calmon

Varia

Médard

et al. 2013

2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Abstract-Lower bounds for the average probability of error of estimating a hidden variable X given an observation of a correlated random variable Y , and Fano's inequality in particular, play a central role in information theory. In this paper, we present a lower bound for the average estimation error based on the marginal distribution of X and the principal inertias of the joint distribution matrix of X and Y . Furthermore, we discuss an information measure based on the sum of the largest principal inertias, called k-correlation, which generalizes maximal correlation. We show that k-correlation satisfies the Data Processing Inequality and is convex in the conditional distribution of Y given X. Finally, we investigate how to answer a fundamental question in inference and privacy: given an observation Y , can we estimate a function f (X) of the hidden random variable X with an average error below a certain threshold? We provide a general method for answering this question using an approach based on rate-distortion theory.

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Section: Remark 2 If λmentioning

confidence: 97%

“…More recently, Kumar and Courtade [16] investigated boolean functions in an information-theoretic context. In particular, they analyzed which is the most informative (in terms of mutual information) 1-bit function (i.e.…”

Section: Introductionmentioning

confidence: 99%