Distributed Hypothesis Testing Under Privacy Constraints
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A distributed binary hypothesis testing problem involving two parties, a remote observer and a detector, is studied. The remote observer has access to a discrete memoryless source, and communicates its observations to the detector via a rate-limited noiseless channel. Having access to another discrete memoryless source, the detector performs hypothesis test on the joint distribution of its own observations with that of the observer. While the goal is to maximize the type 2 error exponent of the test for a given type 1 error probability constraint, it is also desired to keep a private part, which is correlated with the observer's observations, as oblivious to the detector as possible. Considering equivocation and average distortion as the metrics of privacy at the detector, the trade-off between the communication rate of the channel, the type 2 error exponent and privacy is studied. For the general hypothesis testing problem, we establish a single-letter inner bound on the rate-error exponent-equivocation and rate-error exponent-distortion trade-off. Subsequently, a tight single-letter characterization of the rate-error exponent-equivocation and rate-error exponent-distortion trade-off is obtained (i) when the communication rate constraint of the channel is zero, and (ii) for the special case of testing against conditional independence of the observer's observations with that of the detector, given some additional side-information at the detector. Furthermore, the connection between testing against independence problem under a privacy constraint with the information bottleneck and privacy funnel problems are explored. Finally, a strong converse result is proved which shows that the rate-error exponent-equivocation and rate-error exponent-distortion trade-off is independent of the type 1 error probability constraint.
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