Testing Against Conditional Independence Under Security Constraints

Abstract

A distributed binary hypothesis testing problem involving three parties, a remote node, called the observer, a legitimate decoder, called the detector, and an adversary, is studied. The remote node observes a discrete memoryless source, and communicates its observations over a rate-limited noiseless public channel to the detector, which tests for the conditional independence of its own observations from that of the remote node, conditioned on some additional side information. The adversary, in addition to observing the public message, has access to its own correlated side-information. Considering the type 2 error exponent for a given type 1 error probability constraint as the performance measure for the hypothesis test at the detector, and equivocation of the source at the adversary as the secrecy measure, a single-letter characterization of the rate-error exponent-equivocation trade-off is established. Additionally, for a general distortion measure, imposing the average distortion at the adversary as the measure of secrecy achieved, an inner bound on the trade-off between the rate, error exponent and average distortion is obtained. This bound is shown to be tight under the less noisy condition on the adversary's side information.

Testing Against Independence with an Eavesdropper

“…Remark 1: In the limit ǫ → 0 and for ∆ 1 ≥ H Q (X|Z), the fundamental rate-exponent-equivocations region in Theorem 1 recovers the regions presented in [8], [9], which only considered an equivocation constraint under the null hypothesis H 0 . Exponent in [8] Exponent in [1] Fig.…”

“…As we show, in the special case of testing against independence, the type-II error exponents proposed in [8], [9] are optimal in the limit of vanishing type-I error probabilities ǫ → 0 but are generally suboptimal for fixed ǫ > 0. For general ǫ > 0, the optimal exponent is achieved by using the scheme in [8], [9] with probability (1−ǫ) and using a degenerate scheme with probability ǫ. In this degenerate scheme, Alice sends a dummy zero-message, and upon receiving this message, Bob declares the alternative hypothesis H 1 .…”

“…The secrecy scenario with an external eavesdropper that we study in the present paper, was already treated in [8] and in [9] for the more general scenario of testing against conditional independence. As we show, in the special case of testing against independence, the type-II error exponents proposed in [8], [9] are optimal in the limit of vanishing type-I error probabilities ǫ → 0 but are generally suboptimal for fixed ǫ > 0. For general ǫ > 0, the optimal exponent is achieved by using the scheme in [8], [9] with probability (1−ǫ) and using a degenerate scheme with probability ǫ.…”

Distributed Hypothesis Testing over a Noisy Channel: Error-Exponents Trade-Off

“…In the recent years, there has been a renewed interest in distributed statistical inference problems motivated by emerging machine learning applications to be served at the wireless edge, particularly in the context of semantic communications in 5G/6G communication systems [15,16]. Several extensions of the DHT over a noiseless channel problem have been studied, such as generalizations to multi-terminal settings [9,[17][18][19][20][21], DHT under security or privacy constraints [22][23][24][25], DHT with lossy compression [26], interactive settings [27,28], successive refinement models [29], and more. Improved bounds have been obtained on the type I and type II error-exponents region [30,31], and on κ se ( ) for testing correlation between bivariate standard normal distributions [32].…”

Distributed Hypothesis Testing over a Noisy Channel: Error-exponents Trade-off

“…Recently, there is a renewed interest in the distributed HT problem and extensions of the problem has been studied in several different contexts, e.g. multi-terminal settings [10], [11], [12], [13], under security or privacy constraints [14], [15] [16] [17], along with lossy compression [18], etc. to name a few.…”