Distributed Hypothesis Testing With Variable-Length Coding

Abstract

The problem of distributed testing against independence with variable-length coding is considered when the average and not the maximum communication load is constrained as in previous works. The paper characterizes the optimum type-II error exponent of a single-sensor single-decision center system given a maximum type-I error probability when communication is either over a noise-free rate-R link or over a noisy discrete memoryless channel (DMC) with stop-feedback. Specifically, let denote the maximum allowed type-I error probability. Then the optimum exponent of the system with a rate-R link under a constraint on the average communication load coincides with the optimum exponent of such a system with a rate R/(1 −) link under a maximum communication load constraint. A strong converse thus does not hold under an average communication load constraint. A similar observation also holds for testing against independence over DMCs. With variable-length coding and stop-feedback and under an average communication load constraint, the optimum type-II error exponent over a DMC of capacity C equals the optimum exponent under fixed-length coding and a maximum communication load constraint when communication is over a DMC of capacity C(1 −) −1 .

A Fundamental Limit of Distributed Hypothesis Testing Under Memoryless Quantization

“…A recent strand of works relax the "hard" communication constraints and study the dependence testing problem by limiting the expected number of bits sent. A partial list of the studies adopting this perspective is [27]- [32]. The current study is also in line with this perspective.…”

Optimal Exponents in Cascaded Hypothesis Testing under Expected Rate Constraints

“…Since by [8], [11], ∀i ∈ {1, 2} and ∀j ∈ {2, 3} the probabilities Pr[ HY,i = 1|H = 0], and Pr[ HZ,j = 1|H = 0] tend to 0 as n → ∞, we conclude that for the general scheme in Section IV lim n→∞ α 1,n ≤ 1 and lim n→∞ α 2,n ≤ 2 .…”

“…While most information-theoretic works on distributed hypothesis testing constrain the maximum communication rates between the terminals [1]- [6], some recent works [7]- [10] have considered expected rate constraints. Expected rate constraints were first considered in [7], [8] in a single-sensor single-decision center setup, and the maximum error exponents were exactly characterized for testing-against independence when under the alternative hypothesis, the observations are distributed according to the product of the distributions under the null hypothesis. The optimal error exponent for this setup [7], [8], is achieved by a simple coding and decision scheme which chooses an event S n of probability close to the permissible type-I error probability .…”

“…Expected rate constraints were first considered in [7], [8] in a single-sensor single-decision center setup, and the maximum error exponents were exactly characterized for testing-against independence when under the alternative hypothesis, the observations are distributed according to the product of the distributions under the null hypothesis. The optimal error exponent for this setup [7], [8], is achieved by a simple coding and decision scheme which chooses an event S n of probability close to the permissible type-I error probability . Under this event, the transmitter sends a single bit to the decision center, allowing it to decide directly on the hypothesis H = 1.…”

“…Notation: We follow the notation in [13], [8]. In particular, we use sans serif font for bit-strings: e.g., m for a deterministic and M for a random bit-string.…”