We consider the problem of sensor selection for time-optimal detection of a hypothesis. We consider a group of sensors transmitting their observations to a fusion center. The fusion center considers the output of only one randomly chosen sensor at the time, and performs a sequential hypothesis test. We consider the class of sequential tests which are easy to implement, asymptotically optimal, and computationally amenable. For three distinct performance metrics, we show that, for a generic set of sensors and binary hypothesis, the fusion center needs to consider at most two sensors. We also show that for the case of multiple hypothesis, the optimal policy needs at most as many sensors to be observed as the number of underlying hypotheses.
This paper considers the data association problem for multi-target tracking. Multiple hypothesis tracking is a popular algorithm for solving this problem but it is NP-hard and is is quite complicated for a large number of targets or for tracking maneuvering targets. To improve tracking performance and enhance robustness, we propose a randomized multiple model multiple hypothesis tracking method, which has three distinctive advantages. First, it yields a randomized data association solution which maximizes the expectation of the logarithm of the posterior probability and can be solved efficient…
The maximum type-I and type-II error exponents associated with the newly introduced almost-fixed-length hypothesis testing is characterized. In this class of tests, the decision-maker declares the true hypothesis almost always after collecting a fixed number of samples $n$; however in very rare cases with exponentially small probability the decision maker is allowed to collect another set of samples (no more than polynomial in $n$). This class of hypothesis tests are shown to bridge the gap between the classical hypothesis testing with a fixed sample size and the sequential hypothesis testing,…
We present a simple randomized procedure for the prediction of a binary sequence. The algorithm uses ideas from recent developments of the theory of the prediction of individual sequences. We show that if the sequence is a realization of a stationary and ergodic random process then the average number of mistakes converges, almost surely, to that of the optimum, given by the Bayes predictor. The desirable finite-sample properties of the predictor are illustrated by its performance for Markov processes. In such cases the predictor exhibits near optimal behavior even without knowing the order of …
Posterior matching (PM) is a sequential horizon-free feedback communication scheme introduced by the authors, who also provided a rather involved optimality proof showing it achieves capacity for a large class of memoryless channels. Naghshvar et al considered a non-sequential variation of PM with a fixed number of messages and a random decision-time, and gave a simpler proof establishing its optimality via a novel Shannon-Jensen divergence argument. Another simpler optimality proof was given by Li and El Gamal, who considered a fixed-rate fixed block-length variation of PM with an additional …
In this paper, we consider sequential testing over a single-sensor, a single-decision center setup. At each time instant $t$, the sensor gets $k$ samples $(k>0)$ and describes the observed sequence until time $t$ to the decision center over a zero-rate noiseless link. The decision center sends a single bit of feedback to the sensor to request for more samples for compression/testing or to stop the transmission. We have characterized the optimal exponent of type-II error probability under the constraint that type-I error probability does not exceed a given threshold $ε\in (0,1)$ and also when t…
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 hy…
We consider nonparametric sequential hypothesis testing problem when the distribution under the null hypothesis is fully known but the alternate hypothesis corresponds to some other unknown distribution with some loose constraints. We propose a simple algorithm to address the problem. These problems are primarily motivated from wireless sensor networks and spectrum sensing in Cognitive Radios. A decentralized version utilizing spatial diversity is also proposed. Its performance is analysed and asymptotic properties are proved. The simulated and analysed performance of the algorithm is compared…
We consider the problem of decentralized sequential active hypothesis testing (DSAHT), where two transmitting agents, each possessing a private message, are actively helping a third agent--and each other--to learn the message pair over a discrete memoryless multiple access channel (DM-MAC). The third agent (receiver) observes the noisy channel output, which is also available to the transmitting agents via noiseless feedback. We formulate this problem as a decentralized dynamic team, show that optimal transmission policies have a time-invariant domain, and characterize the solution through a dy…
We consider the problem of sequential binary hypothesis testing with a distributed sensor network in a non-Gaussian noise environment. To this end, we present a general formulation of the Consensus + Innovations Sequential Probability Ratio Test (CISPRT). Furthermore, we introduce two different concepts for robustifying the CISPRT and propose four different algorithms, namely, the Least-Favorable-Density-CISPRT, the Median-CISPRT, the M-CISPRT, and the Myriad-CISPRT. Subsequently, we analyze their suitability for different binary hypothesis tests before verifying and evaluating their performan…
We consider a problem of simple hypothesis testing using a randomized test via a tunable loss function proposed by Liao \textit{et al}. In this problem, we derive results that correspond to the Neyman--Pearson lemma, the Chernoff--Stein lemma, and the Chernoff-information in the classical hypothesis testing problem. Specifically, we prove that the optimal error exponent of our problem in the Neyman--Pearson's setting is consistent with the classical result. Moreover, we provide lower bounds of the optimal Bayesian error exponent.
In this paper, we consider a novel $M$-ary sequential hypothesis testing problem in which an adversary is present and perturbs the distributions of the samples before the decision maker observes them. This problem is formulated as a sequential adversarial hypothesis testing game played between the decision maker and the adversary. This game is a zero-sum and strategic one. We assume the adversary is active under \emph{all} hypotheses and knows the underlying distribution of observed samples. We adopt this framework as it is the worst-case scenario from the perspective of the decision maker. Th…
We consider nonparametric or universal sequential hypothesis testing problem when the distribution under the null hypothesis is fully known but the alternate hypothesis corresponds to some other unknown distribution. These algorithms are primarily motivated from spectrum sensing in Cognitive Radios and intruder detection in wireless sensor networks. We use easily implementable universal lossless source codes to propose simple algorithms for such a setup. The algorithms are first proposed for discrete alphabet. Their performance and asymptotic properties are studied theoretically. Later these a…
We consider a multi-hypothesis testing problem involving a K-armed bandit. Each arm's signal follows a distribution from a vector exponential family. The actual parameters of the arms are unknown to the decision maker. The decision maker incurs a delay cost for delay until a decision and a switching cost whenever he switches from one arm to another. His goal is to minimise the overall cost until a decision is reached on the true hypothesis. Of interest are policies that satisfy a given constraint on the probability of false detection. This is a sequential decision making problem where the deci…
Consider a decision maker who is responsible to collect observations so as to enhance his information in a speedy manner about an underlying phenomena of interest. The policies under which the decision maker selects sensing actions can be categorized based on the following two factors: i) sequential vs. non-sequential; ii) adaptive vs. non-adaptive. Non-sequential policies collect a fixed number of observation samples and make the final decision afterwards; while under sequential policies, the sample size is not known initially and is determined by the observation outcomes. Under adaptive poli…
This work investigates the sequential hypothesis testing problem with online sensor selection and sensor usage constraints. That is, in a sensor network, the fusion center sequentially acquires samples by selecting one "most informative" sensor at each time until a reliable decision can be made. In particular, the sensor selection is carried out in the online fashion since it depends on all the previous samples at each time. Our goal is to develop the sequential test (i.e., stopping rule and decision function) and sensor selection strategy that minimize the expected sample size subject to the …
We consider the classical sequential binary hypothesis testing problem in which there are two hypotheses governed respectively by distributions $P_0$ and $P_1$ and we would like to decide which hypothesis is true using a sequential test. It is known from the work of Wald and Wolfowitz that as the expectation of the length of the test grows, the optimal type-I and type-II error exponents approach the relative entropies $D(P_1\|P_0)$ and $D(P_0\|P_1)$. We refine this result by considering the optimal backoff---or second-order asymptotics---from the corner point of the achievable exponent region …
Hypothesis testing is an important problem with applications in target localization, clinical trials etc. Many active hypothesis testing strategies operate in two phases: an exploration phase and a verification phase. In the exploration phase, selection of experiments is such that a moderate level of confidence on the true hypothesis is achieved. Subsequent experiment design aims at improving the confidence level on this hypothesis to the desired level. In this paper, the focus is on the verification phase. A confidence measure is defined and active hypothesis testing is formulated as a confid…
We investigate the quantize and binning scheme, known as the Shimokawa-Han-Amari (SHA) scheme, for the distributed hypothesis testing. We develop tools to evaluate the critical rate attainable by the SHA scheme. For a product of binary symmetric double sources, we present a sequential scheme that improves upon the SHA scheme.
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 $ε$, we obtain certain fundamental limits on the asymptotic behavior of the sequential test as $n$ tends to infinity. Assuming that $Γ$ is a convex an…
This work analyzes the asymptotic performances of fully distributed sequential hypothesis testing procedures as the type-I and type-II error rates approach zero, in the context of a sensor network without a fusion center. In particular, the sensor network is defined by an undirected graph, where each sensor can observe samples over time, access the information from the adjacent sensors, and perform the sequential test based on its own decision statistic. Different from most literature, the sampling process and the information exchange process in our framework take place simultaneously (or, at …
We provide a novel analysis of Wald's sequential probability ratio test based on information theoretic measures for symmetric thresholds, symmetric noise, and equally likely hypotheses under the assumption that the test exactly terminates at one of the thresholds. This test is optimal in the sense that it yields the minimum mean decision time. To analyze the decision-making process we consider information densities, which represent the stochastic information content of the observations yielding a stochastic termination time of the test. Based on this, we show that the conditional probability t…
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 relat…
A sequential problem in decentralized detection is considered. Two observers can make repeated noisy observations of a binary hypothesis on the state of the environment. At any time, observer 1 can stop and send a final binary message to observer 2 or it may continue to take more measurements. Every time observer 1 postpones its final message to observer 2, it incurs a penalty. Observer 2's operation under two different scenarios is explored. In the first scenario, observer 2 waits to receive the final message from observer 1 and then starts taking measurements of its own. It is then faced wit…
This paper considers the sequential composite hypothesis test with multiple sensors. The sensors observe random samples in parallel and communicate with a fusion center, who makes the global decision based on the sensor inputs. On one hand, in the centralized scenario, where local samples are precisely transmitted to the fusion center, the generalized sequential likelihood ratio test (GSPRT) is shown to be asymptotically optimal in terms of the expected sample size as error rates tend to zero. On the other hand, for systems with limited power and bandwidth resources, decentralized solutions th…
School of Electrical Engineering, Tel-Aviv University（特拉维夫大学电气工程学院）；Dept. of EECS, MIT（麻省理工学院电气工程与计算机科学系）
We consider the question of sequential prediction under the log-loss in terms of cumulative regret. Namely, given a hypothesis class of distributions, learner sequentially predicts the (distribution of the) next letter in sequence and its performance is compared to the baseline of the best constant predictor from the hypothesis class. The well-specified case corresponds to an additional assumption that the data-generating distribution belongs to the hypothesis class as well. Here we present results in the more general misspecified case. Due to special properties of the log-loss, the same probl…
