Reliability of sequential hypothesis testing can be achieved by an almost-fixed-length test

Lalitha, Anusha; Javidi, Tara

doi:10.1109/isit.2016.7541591

Cited by 15 publications

(18 citation statements)

References 10 publications

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“…where (55) (30), (49) and (50). Note that (57) holds with equality if and only if P X is equiprobable on f −1 (y) for every y ∈ Y such that P X (f −1 (y)) > 0.…”

Section: A Upper Boundsmentioning

confidence: 99%

Arimoto–Rényi Conditional Entropy and Bayesian $M$ -Ary Hypothesis Testing

Sason

Verdú

2018

IEEE Trans. Inform. Theory

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This paper gives upper and lower bounds on the minimum error probability of Bayesian Mary hypothesis testing in terms of the Arimoto-Rényi conditional entropy of an arbitrary order α. The improved tightness of these bounds over their specialized versions with the Shannon conditional entropy (α = 1) is demonstrated. In particular, in the case where M is finite, we show how to generalize Fano's inequality under both the conventional and list-decision settings. As a counterpart to the generalized Fano's inequality, allowing M to be infinite, a lower bound on the Arimoto-Rényi conditional entropy is derived as a function of the minimum error probability. Explicit upper and lower bounds on the minimum error probability are obtained as a function of the Arimoto-Rényi conditional entropy for both positive and negative α. Furthermore, we give upper bounds on the minimum error probability as functions of the Rényi divergence. In the setup of discrete memoryless channels, we analyze the exponentially vanishing decay of the Arimoto-Rényi conditional entropy of the transmitted codeword given the channel output when averaged over a random-coding ensemble. Index TermsArimoto-Rényi conditional entropy, Bayesian minimum probability of error, Chernoff information, Fano's inequality, list decoding, M -ary hypothesis testing, random coding, Rényi entropy, Rényi divergence. I. Sason is with the Andrew and Erna Viterbi

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“…where (55) (30), (49) and (50). Note that (57) holds with equality if and only if P X is equiprobable on f −1 (y) for every y ∈ Y such that P X (f −1 (y)) > 0.…”

Section: A Upper Boundsmentioning

confidence: 99%

Arimoto–Rényi Conditional Entropy and Bayesian $M$ -Ary Hypothesis Testing

Sason

Verdú

2018

IEEE Trans. Inform. Theory

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“…In other words, the average number of copies used in the testing procedure should be bounded by some number n. The second type of constraint is the probabilistic constraint [26,27] max i∈{0,1}…”

Section: Constraints and Achievable Exponentsmentioning

confidence: 99%

Optimal Adaptive Strategies for Sequential Quantum Hypothesis Testing

Li,

Tan,

Tomamichel

2021

Preprint

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We consider sequential hypothesis testing between two quantum states using adaptive and non-adaptive strategies. In this setting, samples of an unknown state are requested sequentially and a decision to either continue or to accept one of the two hypotheses is made after each test. Under the constraint that the number of samples is bounded, either in expectation or with high probability, we exhibit adaptive strategies that minimize both types of misidentification errors. Namely, we show that these errors decrease exponentially (in the stopping time) with decay rates given by the measured relative entropies between the two states. Moreover, if we allow joint measurements on multiple samples, the rates are increased to the respective quantum relative entropies. We also fully characterize the achievable error exponents for non-adaptive strategies and provide numerical evidence showing that adaptive measurements are necessary to achieve our bounds under some additional assumptions.

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“…In addition, the corner point (D(p 1 p 0 ), D(p 0 p 1 )) is attained by a sequence of sequential probability ratio tests (SPRTs). Lalitha and Javidi [6] considered an interesting setting that interpolates between the fixed-length hypothesis testing and sequential hypothesis testing. They considered the almost-fixed-length hypothesis testing problem, in which the stopping time is allowed to be larger than a prescribed integer n with exponentially small probability exp(−nγ) for different γ > 0.…”

Section: A Related Workmentioning

confidence: 99%

“…They considered the almost-fixed-length hypothesis testing problem, in which the stopping time is allowed to be larger than a prescribed integer n with exponentially small probability exp(−nγ) for different γ > 0. The probabilistic constraints we employ in this paper are analogous to those in [6], but instead of allowing the event that the stopping time to be larger than n to have exponentially small probability, we only require this event to have probability at most ∈ (0, 1), a fixed constant. This allows us to ask questions ranging from strong converses to second-order asymptotics.…”

Section: A Related Workmentioning

confidence: 99%

Asymptotics of Sequential Composite Hypothesis Testing under Probabilistic Constraints

Pan¹,

Lü²,

Tan³

2021

Preprint

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We consider the sequential composite binary hypothesis testing problem in which one of the hypotheses is governed by a single distribution while the other is governed by a family of distributions whose parameters belong to a known set Γ. We would like to design a test to decide which hypothesis is in effect. Under the constraints that the probabilities that the length of the test, a stopping time, exceeds n are bounded by a certain threshold , we obtain certain fundamental limits on the asymptotic behavior of the sequential test as n tends to infinity. Assuming that Γ is a convex and compact set, we obtain the set of all first-order error exponents for the problem. We also prove a strong converse. Additionally, we obtain the set of second-order error exponents under the assumption that X is a finite alphabet. In the proof of second-order asymptotics, a main technical contribution is the derivation of a central limit-type result for a maximum of an uncountable set of log-likelihood ratios under suitable conditions. This result may be of independent interest. We also show that some important statistical models satisfy the conditions.

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Reliability of sequential hypothesis testing can be achieved by an almost-fixed-length test

Cited by 15 publications

References 10 publications

Arimoto–Rényi Conditional Entropy and Bayesian $M$ -Ary Hypothesis Testing

Arimoto–Rényi Conditional Entropy and Bayesian $M$ -Ary Hypothesis Testing

Optimal Adaptive Strategies for Sequential Quantum Hypothesis Testing

Asymptotics of Sequential Composite Hypothesis Testing under Probabilistic Constraints

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