Sequential Hypothesis Testing with Broadcast Failures (original) (raw)

Hypothesis Testing in Feedforward Networks With Broadcast Failures

IEEE Journal of Selected Topics in Signal Processing, 2000

Consider a large number of nodes, which sequentially make decisions between two given hypotheses. Each node takes a measurement of the underlying truth, observes the decisions from some immediate predecessors, and makes a decision between the given hypotheses. We consider two classes of broadcast failures: 1) each node broadcasts a decision to the other nodes, subject to random erasure in the form of a binary erasure channel; 2) each node broadcasts a randomly flipped decision to the other nodes in the form of a binary symmetric channel. We are interested in conditions under which there does (or does not) exist a decision strategy consisting of a sequence of likelihood ratio tests such that the node decisions converge in probability to the underlying truth, as the number of nodes goes to infinity. In both cases, we show that if each node only learns from a bounded number of immediate predecessors, then there does not exist a decision strategy such that the decisions converge in probability to the underlying truth. However, in case 1, we show that if each node learns from an unboundedly growing number of predecessors, then there exists a decision strategy such that the decisions converge in probability to the underlying truth, even when the erasure probabilities converge to 1. We show that a locally optimal strategy, consisting of a sequence of Bayesian likelihood ratio tests, is such a strategy, and we derive the convergence rate of the error probability for this strategy. In case 2, we show that if each node learns from all of its previous predecessors, then there exists a decision strategy such that the decisions converge in probability to the underlying truth when the flipping probabilities of the binary symmetric channels are bounded away from 1/2. Again, we show that a locally optimal strategy achieves this, and we derive the convergence rate of the error probability for it. In the case where the flipping probabilities converge to 1/2, we derive a necessary condition on the convergence rate of the flipping probabilities such that the decisions based on the locally optimal strategy still converge to the underlying truth. We also explicitly characterize the relationship between the convergence rate of the error probability and the convergence rate of the flipping probabilities.

Novel algorithms for distributed sequential hypothesis testing

2011

This paper considers sequential hypothesis testing in a decentralized framework. We start with two simple decentralized sequential hypothesis testing algorithms. One of which is later proved to be asymptotically Bayes optimal. We also consider composite versions of decentralized sequential hypothesis testing. A novel nonparametric version for decentralized sequential hypothesis testing using universal source coding theory is developed. Finally we design a simple decentralized multihypothesis sequential detection algorithm.

Decentralized Sequential Hypothesis Testing Using Asynchronous Communication

IEEE Transactions on Information Theory, 2000

We present a test for the problem of decentralized sequential hypothesis testing, which is asymptotically optimum. By selecting a suitable sampling mechanism at each sensor, communication between sensors and fusion center is asynchronous and limited to 1-bit data. The proposed SPRT-like test turns out to be order-2 asymptotically optimum in the case of continuous time and continuous path signals, while in discrete time this strong asymptotic optimality property is preserved under proper conditions. If these conditions do not hold, then we can show optimality of order-1. Simulations corroborate the excellent performance characteristics of the test of interest.

Asymptotically optimum tests for decentralized sequential testing in continuous time

We propose an asymptotically optimum test for the problem of decentralized sequential hypothesis testing in continuous time, in the case where the sensors have full local memory and no feedback from the fusion center. According to our scheme, the sensors perform locally repeated SPRTs and communicate, asynchronously, their one-bit decisions to the fusion center. The fusion center in turn uses the received information to perform a centralized SPRT in order to make the final decision. The expected time for a decision of the proposed scheme differs from the optimum continuous-time centralized SPRT only by a constant. This fact suggests order-2 asymptotic optimality of our test as compared to existing schemes that are optimal of order-1. Moreover, simulation experiments reveal that the performance of our scheme is significantly better than that of the discrete-time centralized SPRT.

Distributed binary hypothesis testing with feedback

IEEE Transactions on Systems, Man, and Cybernetics, 1995

The problem of binary hypothesis testing is revisited in the context of distributed detection with feedback. Two basic distributed structures with decision feedback are considered. The first structure is the fusion center network, with decision feedback connections from the fusion center element to each one of the subordinate decisionmakers. The second structure consists of a set of detectors that are fully interconnected via decision feedback. Both structures are optimized in the Neyman-Pearson sense by optimizing each decisionmaker individually. Then, the time evolution of the power of the tests is derived. Definite conclusions regarding the gain induced by the feedback process and direct comparisons between the two structures and the optimal centralized scheme are obtained through asymptotic studies (that is, assuming the presence of asymptotically many local detectors). The behavior of these structures is also examined in the presence of variations in the statistical description of the hypotheses. Specific robust designs are proposed and the benefits from robust operations are established. Numerical results provide additional support to the theoretical arguments.

Unstructured sequential testing in sensor networks

52nd IEEE Conference on Decision and Control, 2013

We consider the problem of quickly detecting a signal in a sensor network when the subset of sensors in which signal may be present is completely unknown. We formulate this problem as a sequential hypothesis testing problem with a simple null (signal is absent everywhere) and a composite alternative (signal is present somewhere). We introduce a novel class of scalable sequential tests which, for any subset of affected sensors, minimize the expected sample size for a decision asymptotically, that is as the error probabilities go to 0. Moreover, we propose sequential tests that require minimal transmission activity from the sensors to the fusion center, while preserving this asymptotic optimality property.

Sequential hypothesis tests under random horizon

Sequential Analysis, 2020

We consider a problem of sequential testing a simple hypothesis against a simple alternative, based on observations of a discrete-time stochastic process X 1 , X 2 , :::, in the presence of a random horizon H. At any time n of the experiment, the statistician is only informed whether H > n or not. In this latter case, the experiment should be terminated and the final decision on the acceptance or rejection of the hypothesis should be taken on the basis of the available observations X 1 , X 2 , :::, X n (n ¼ 1, 2, :::). H is assumed to be independent of the observations, and its distribution is known to the statistician. Under the random horizon, we consider a variant of the modified Kiefer-Weiss problem: given restrictions on the probabilities of errors, minimize the average sample size calculated under the assumption that the observations follow a fixed distribution, not necessarily one of those hypothesized. Under suitable conditions on the process and/or the horizon, we characterize the structure of all optimal sequential tests in this problem. Then, we apply these results to characterize optimal tests in the case of independent observations. On the basis of the general theory, more specific results are obtained for independent and identically distributed (i.i.d.) observations with a geometrically distributed horizon. In a simple sampling model, we solve the Kiefer-Weiss problem under the random horizon model. We also discuss the questions of Wald-Wolfowitz optimality in the presence of the random horizon. In particular, we show that the stopping rules of the optimal tests, minimizing the average sample size under one of the hypotheses, are randomized versions of those of Wald's sequential probability ratio tests.

Distributed M-ary hypothesis testing with binary local decisions

Information Fusion, 2004

Parallel distributed detection schemes for M-ary hypothesis testing often assume that for each observation the local detector transmits at least log 2 M bits to a data fusion center (DFC). However, it is possible for less than log 2 M bits to be available, and in this study we consider 1-bit local detectors with M > 2. We develop conditions for asymptotic detection of the correct hypothesis by the DFC, formulate the optimal decision rules for the DFC, and derive expressions for the performance of the system. Local detector design is demonstrated in examples, using genetic algorithm search for local decision thresholds. We also provide an intuitive geometric interpretation for the partitioning of the observations into decision regions. The interpretation is presented in terms of the joint probability of the local decisions and the hypotheses.

Simple CHT: A new derivation of the weakest failure detector for consensus

The paper proposes an alternative proof that Ω, an oracle that outputs a process identifier and guarantees that eventually the same correct process identifier is output at all correct processes, provides minimal information about failures for solving consensus in read-write shared-memory systems: every oracle that gives enough failure information to solve consensus can be used to implement Ω.