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Probabilistic Methods in Information Theory, Hypothesis Testing, and Coding

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Information Theory, Probability and Statistics".

Deadline for manuscript submissions: closed (31 August 2019) | Viewed by 22479

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Special Issue Editor

Prof. Dr. Igal Sason

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Guest Editor

Andrew and Erna Viterbi Faculty of Electrical Engineering, Technion–Israel Institute of Technology, Haifa 32000, Israel
Interests: information theory; coding theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Probabilistic methods play a key role in establishing direct and converse results in information theory, statistical hypothesis testing and coding. In this Special Issue, we welcome unpublished contributions related to such probabilistic tools and their information and coding-theoretic applications. Examples include probabilistic methods which are used to establish results in channel coding, lossless and lossy source coding such as concentration of measure inequalities, large deviations, method of types, martingales, majorization theory, coupling and Stein's method.

Prof. Dr. Igal Sason
Guest Editor

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Keywords

channel coding
data compression
universal coding
hypothesis testing
information measures
method of types
concentration of measures
martingales
Stein's method

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Published Papers (6 papers)

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Research

26 pages, 544 KiB

Open AccessArticle

Guessing with a Bit of Help

by Nir Weinberger and Ofer Shayevitz

Entropy 2020, 22(1), 39; https://doi.org/10.3390/e22010039 - 26 Dec 2019

Cited by 5 | Viewed by 3312

Abstract

What is the value of just a few bits to a guesser? We study this problem in a setup where Alice wishes to guess an independent and identically distributed (i.i.d.) random vector and can procure a fixed number of k information bits from Bob, who has observed this vector through a memoryless channel. We are interested in the guessing ratio, which we define as the ratio of Alice’s guessing-moments with and without observing Bob’s bits. For the case of a uniform binary vector observed through a binary symmetric channel, we provide two upper bounds on the guessing ratio by analyzing the performance of the dictator (for general

k \geq 1

) and majority functions (for

k = 1

). We further provide a lower bound via maximum entropy (for general

k \geq 1

) and a lower bound based on Fourier-analytic/hypercontractivity arguments (for

k = 1

). We then extend our maximum entropy argument to give a lower bound on the guessing ratio for a general channel with a binary uniform input that is expressed using the strong data-processing inequality constant of the reverse channel. We compute this bound for the binary erasure channel and conjecture that greedy dictator functions achieve the optimal guessing ratio. Full article

(This article belongs to the Special Issue Probabilistic Methods in Information Theory, Hypothesis Testing, and Coding)

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25 pages, 903 KiB

Open AccessArticle

Conditional Rényi Divergence Saddlepoint and the Maximization of α-Mutual Information

by Changxiao Cai and Sergio Verdú

Entropy 2019, 21(10), 969; https://doi.org/10.3390/e21100969 - 4 Oct 2019

Cited by 13 | Viewed by 3594

Abstract

Rényi-type generalizations of entropy, relative entropy and mutual information have found numerous applications throughout information theory and beyond. While there is consensus that the ways A. Rényi generalized entropy and relative entropy in 1961 are the “right” ones, several candidates have been put [...] Read more.

α

. In this paper we lend further evidence to the notion that a Bayesian measure of statistical distinctness introduced by R. Sibson in 1969 (closely related to Gallager’s

E_{0}

function) is the most natural generalization, lending itself to explicit computation and maximization, as well as closed-form formulas. This paper considers general (not necessarily discrete) alphabets and extends the major analytical results on the saddle-point and saddle-level of the conditional relative entropy to the conditional Rényi divergence. Several examples illustrate the main application of these results, namely, the maximization of

α

-mutual information with and without constraints. Full article

(This article belongs to the Special Issue Probabilistic Methods in Information Theory, Hypothesis Testing, and Coding)

27 pages, 432 KiB

Open AccessEditor’s ChoiceArticle

Distributed Hypothesis Testing with Privacy Constraints

by Atefeh Gilani, Selma Belhadj Amor, Sadaf Salehkalaibar and Vincent Y. F. Tan

Entropy 2019, 21(5), 478; https://doi.org/10.3390/e21050478 - 7 May 2019

Cited by 16 | Viewed by 3857

Abstract

We revisit the distributed hypothesis testing (or hypothesis testing with communication constraints) problem from the viewpoint of privacy. Instead of observing the raw data directly, the transmitter observes a sanitized or randomized version of it. We impose an upper bound on the mutual information between the raw and randomized data. Under this scenario, the receiver, which is also provided with side information, is required to make a decision on whether the null or alternative hypothesis is in effect. We first provide a general lower bound on the type-II exponent for an arbitrary pair of hypotheses. Next, we show that if the distribution under the alternative hypothesis is the product of the marginals of the distribution under the null (i.e., testing against independence), then the exponent is known exactly. Moreover, we show that the strong converse property holds. Using ideas from Euclidean information theory, we also provide an approximate expression for the exponent when the communication rate is low and the privacy level is high. Finally, we illustrate our results with a binary and a Gaussian example. Full article

(This article belongs to the Special Issue Probabilistic Methods in Information Theory, Hypothesis Testing, and Coding)

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15 pages, 361 KiB

Open AccessArticle

Guessing with Distributed Encoders

by Annina Bracher, Amos Lapidoth and Christoph Pfister

Entropy 2019, 21(3), 298; https://doi.org/10.3390/e21030298 - 19 Mar 2019

Cited by 7 | Viewed by 2886

Abstract

Two correlated sources emit a pair of sequences, each of which is observed by a different encoder. Each encoder produces a rate-limited description of the sequence it observes, and the two descriptions are presented to a guessing device that repeatedly produces sequence pairs until correct. The number of guesses until correct is random, and it is required that it have a moment (of some prespecified order) that tends to one as the length of the sequences tends to infinity. The description rate pairs that allow this are characterized in terms of the Rényi entropy and the Arimoto–Rényi conditional entropy of the joint law of the sources. This solves the guessing analog of the Slepian–Wolf distributed source-coding problem. The achievability is based on random binning, which is analyzed using a technique by Rosenthal. Full article

(This article belongs to the Special Issue Probabilistic Methods in Information Theory, Hypothesis Testing, and Coding)

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31 pages, 995 KiB

Open AccessArticle

Detection Games under Fully Active Adversaries

by Benedetta Tondi, Neri Merhav and Mauro Barni

Entropy 2019, 21(1), 23; https://doi.org/10.3390/e21010023 - 29 Dec 2018

Cited by 8 | Viewed by 3039

Abstract

We study a binary hypothesis testing problem in which a defender must decide whether a test sequence has been drawn from a given memoryless source

P_{0}

, while an attacker strives to impede the correct detection. With respect to previous works, the [...] Read more.

We study a binary hypothesis testing problem in which a defender must decide whether a test sequence has been drawn from a given memoryless source

P_{0}

, while an attacker strives to impede the correct detection. With respect to previous works, the adversarial setup addressed in this paper considers an attacker who is active under both hypotheses, namely, a fully active attacker, as opposed to a partially active attacker who is active under one hypothesis only. In the fully active setup, the attacker distorts sequences drawn both from

P_{0}

and from an alternative memoryless source

P_{1}

, up to a certain distortion level, which is possibly different under the two hypotheses, to maximize the confusion in distinguishing between the two sources, i.e., to induce both false positive and false negative errors at the detector, also referred to as the defender. We model the defender–attacker interaction as a game and study two versions of this game, the Neyman–Pearson game and the Bayesian game. Our main result is in the characterization of an attack strategy that is asymptotically both dominant (i.e., optimal no matter what the defender’s strategy is) and universal, i.e., independent of

P_{0}

and

P_{1}

. From the analysis of the equilibrium payoff, we also derive the best achievable performance of the defender, by relaxing the requirement on the exponential decay rate of the false positive error probability in the Neyman–Pearson setup and the tradeoff between the error exponents in the Bayesian setup. Such analysis permits characterizing the conditions for the distinguishability of the two sources given the distortion levels. Full article

(This article belongs to the Special Issue Probabilistic Methods in Information Theory, Hypothesis Testing, and Coding)

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25 pages, 452 KiB

Open AccessArticle

Tight Bounds on the Rényi Entropy via Majorization with Applications to Guessing and Compression

by Igal Sason

Entropy 2018, 20(12), 896; https://doi.org/10.3390/e20120896 - 22 Nov 2018

Cited by 27 | Viewed by 5144

Abstract

This paper provides tight bounds on the Rényi entropy of a function of a discrete random variable with a finite number of possible values, where the considered function is not one to one. To that end, a tight lower bound on the Rényi entropy of a discrete random variable with a finite support is derived as a function of the size of the support, and the ratio of the maximal to minimal probability masses. This work was inspired by the recently published paper by Cicalese et al., which is focused on the Shannon entropy, and it strengthens and generalizes the results of that paper to Rényi entropies of arbitrary positive orders. In view of these generalized bounds and the works by Arikan and Campbell, non-asymptotic bounds are derived for guessing moments and lossless data compression of discrete memoryless sources. Full article

(This article belongs to the Special Issue Probabilistic Methods in Information Theory, Hypothesis Testing, and Coding)

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