12 research outputs found
Hypothesis testing via a comparator
This paper investigates the best achievable performance by a hypothesis test satisfying a structural constraint: two functions are computed at two different terminals and the detector consists of a simple comparator verifying whether the functions agree. Such tests arise as part of study of fundamental limits of channel coding, but are also useful in other contexts. A simple expression for the Stein exponent is found and applied to showing a strong converse in the problem of multi-terminal hypothesis testing with rate constraints. Connections to the Gács-Körner common information and to spectral properties of conditional expectation operator are identified. Further tightening of results hinges on finding λ-blocks of minimal weight. Application of Delsarte's linear programming method to this problem is described.Center for Science of Information (Grant Agreement CCF-09-39370
Distributed Hypothesis Testing with Privacy Constraints
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
Rate-Exponent Region for a Class of Distributed Hypothesis Testing Against Conditional Independence Problems
We study a class of -encoder hypothesis testing against conditional
independence problems. Under the criterion that stipulates minimization of the
Type II error subject to a (constant) upper bound on the Type I
error, we characterize the set of encoding rates and exponent for both discrete
memoryless and memoryless vector Gaussian settings. For the DM setting, we
provide a converse proof and show that it is achieved using the
Quantize-Bin-Test scheme of Rahman and Wagner. For the memoryless vector
Gaussian setting, we develop a tight outer bound by means of a technique that
relies on the de Bruijn identity and the properties of Fisher information. In
particular, the result shows that for memoryless vector Gaussian sources the
rate-exponent region is exhausted using the Quantize-Bin-Test scheme with
\textit{Gaussian} test channels; and there is \textit{no} loss in performance
caused by restricting the sensors' encoders not to employ time sharing.
Furthermore, we also study a variant of the problem in which the source, not
necessarily Gaussian, has finite differential entropy and the sensors'
observations noises under the null hypothesis are Gaussian. For this model, our
main result is an upper bound on the exponent-rate function. The bound is shown
to mirror a corresponding explicit lower bound, except that the lower bound
involves the source power (variance) whereas the upper bound has the source
entropy power. Part of the utility of the established bound is for
investigating asymptotic exponent/rates and losses incurred by distributed
detection as function of the number of sensors.Comment: Submitted for publication to the IEEE Transactions of Information
Theory. arXiv admin note: substantial text overlap with arXiv:1904.03028,
arXiv:1811.0393
On the Reliability Function of Distributed Hypothesis Testing Under Optimal Detection
The distributed hypothesis testing problem with full side-information is
studied. The trade-off (reliability function) between the two types of error
exponents under limited rate is studied in the following way. First, the
problem is reduced to the problem of determining the reliability function of
channel codes designed for detection (in analogy to a similar result which
connects the reliability function of distributed lossless compression and
ordinary channel codes). Second, a single-letter random-coding bound based on a
hierarchical ensemble, as well as a single-letter expurgated bound, are derived
for the reliability of channel-detection codes. Both bounds are derived for a
system which employs the optimal detection rule. We conjecture that the
resulting random-coding bound is ensemble-tight, and consequently optimal
within the class of quantization-and-binning schemes
Distributed Hypothesis Testing over a Noisy Channel: Error-exponents Trade-off
A two-terminal distributed binary hypothesis testing (HT) problem over a
noisy channel is studied. The two terminals, called the observer and the
decision maker, each has access to independent and identically distributed
samples, denoted by and , respectively. The observer
communicates to the decision maker over a discrete memoryless channel (DMC),
and the decision maker performs a binary hypothesis test on the joint
probability distribution of based on and
the noisy information received from the observer. The trade-off between the
exponents of the type I and type II error probabilities in HT is investigated.
Two inner bounds are obtained, one using a separation-based scheme that
involves type-based compression and unequal error-protection channel coding,
and the other using a joint scheme that incorporates type-based hybrid coding.
The separation-based scheme is shown to recover the inner bound obtained by Han
and Kobayashi for the special case of a rate-limited noiseless channel, and
also the one obtained by the authors previously for a corner point of the
trade-off. Exact single-letter characterization of the optimal trade-off is
established for the special case of testing for the marginal distribution of
, when is unavailable. Our results imply that a
separation holds in this case, in the sense that the optimal trade-off is
achieved by a scheme that performs independent HT and channel coding. Finally,
we show via an example that the joint scheme achieves a strictly tighter bound
than the separation-based scheme for some points of the error-exponent
trade-off