The value of feedback for decentralized detection in large sensor networks

We consider the decentralized binary hypothesis testing problem in networks with feedback, where some or all of the sensors have access to compressed summaries of other sensors' observations. We study certain two-message feedback architectures, in which every sensor sends two messages to a fusion center, with the second message based on full or partial knowledge of the first messages of the other sensors. Under either a Neyman-Pearson or a Bayesian formulation, we show that the asymptotically optimal (in the limit of a large number of sensors) detection performance (as quantified by error exponents) does not benefit from the feedback messages

Bayesian Design of Tandem Networks for Distributed Detection With Multi-bit Sensor Decisions

We consider the problem of decentralized hypothesis testing under communication constraints in a topology where several peripheral nodes are arranged in tandem. Each node receives an observation and transmits a message to its successor, and the last node then decides which hypothesis is true. We assume that the observations at different nodes are, conditioned on the true hypothesis, independent and the channel between any two successive nodes is considered error-free but rate-constrained. We propose a cyclic numerical design algorithm for the design of nodes using a person-by-person methodology with the minimum expected error probability as a design criterion, where the number of communicated messages is not necessarily equal to the number of hypotheses. The number of peripheral nodes in the proposed method is in principle arbitrary and the information rate constraints are satisfied by quantizing the input of each node. The performance of the proposed method for different information rate constraints, in a binary hypothesis test, is compared to the optimum rate-one solution due to Swaszek and a method proposed by Cover, and it is shown numerically that increasing the channel rate can significantly enhance the performance of the tandem network. Simulation results for $M$-ary hypothesis tests also show that by increasing the channel rates the performance of the tandem network significantly improves

Hypothesis Testing in Feedforward Networks with Broadcast Failures

Consider a countably infinite set 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 whether there exists a decision strategy consisting of a sequence of likelihood ratio tests such that the node decisions converge in probability to the underlying truth. 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 the decisions converge in probability to the underlying truth, even when the erasure probabilities converge to 1. We also derive the convergence rate of the error probability. In case 2, we show that if each node learns from all of its previous predecessors, then the decisions converge in probability to the underlying truth when the flipping probabilities of the binary symmetric channels are bounded away from 1/2. 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 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

Bayesian Detection in Bounded Height Tree Networks

We study the detection performance of large scale sensor networks, configured as trees with bounded height, in which information is progressively compressed as it moves towards the root of the tree. We show that, under a Bayesian formulation, the error probability decays exponentially fast, and we provide bounds for the error exponent. We then focus on the case where the tree has certain symmetry properties. We derive the form of the optimal exponent within a restricted class of easily implementable strategies, as well as optimal strategies within that class. We also find conditions under which (suitably defined) majority rules are optimal. Finally, we provide evidence that in designing a network it is preferable to keep the branching factor small for nodes other than the neighbors of the leaves

On decision making in tandem networks

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009."September 2009." Cataloged from PDF version of thesis.Includes bibliographical references (p. 81-82).We study the convergence of Bayesian learning in a tandem social network. Each agent receives a noisy signal about the underlying state of the world, and observes her predecessor's action before choosing her own. We characterize the conditions under which, as the network grows larger, agents' beliefs converge to the true state of the world. The literature has predominantly focused on the case where the number of possible actions is equal to that of alternative states. We examine the case where agents pick three-valued actions to learn one of two possible states of the world. We focus on myopic strategies, and distinguish between learning in probability and learning almost surely. We show that ternary actions are not sufficient to achieve learning (almost sure or in probability) when the likelihood ratios of the private signals are bounded. When the private signals can be arbitrarily informative (unbounded likelihood ratios), we show that there is learning, in probability. Finally, we report an experimental test of how individuals learn from the behavior of others. We explore sequential decision making in a game of three players, where each decision maker observes her immediate predecessor's binary or ternary action. Our experimental design uses Amazon Mechanical Turk, and is based on a setup with continuous signals, discrete actions and a cutoff elicitation technique introduced in [QK05). We replicate the findings of the experimental economics literature on observational learning in the binary action case and use them as a benchmark. We find that herds are less frequent when subjects use three actions instead of two. In addition, our results suggest that with ternary actions, behavior in the laboratory is less consistent with the predictions of Bayesian behavior than with binary actions.by Manal Dia.M.Eng

On the sub-exponential decay of detection error probabilities in long tandems

Abstract—We consider the problem of Bayesian decentralized binary hypothesis testing in a network of sensors arranged in a tandem. We show that the rate of error probability decay is always subexponential, establishing the validity of a long-standing conjecture. Under the additional assumption of bounded Kullback–Leibler (KL) divergences, we show that for all d>1=2, the error probability is (e), where c is a positive constant. Furthermore, the bound (e), for all d>1, holds under an additional mild condition on the distributions. This latter bound is shown to be tight. Index Terms—Decentralized detection, error exponent, serial network, tandem network. I

Optimal Inference for Distributed Detection

In distributed detection, there does not exist an automatic way of generating optimal decision strategies for non-affine decision functions. Consequently, in a detection problem based on a non-affine decision function, establishing optimality of a given decision strategy, such as a generalized likelihood ratio test, is often difficult or even impossible. In this thesis we develop a novel detection network optimization technique that can be used to determine necessary and sufficient conditions for optimality in distributed detection for which the underlying objective function is monotonic and convex in probabilistic decision strategies. Our developed approach leverages on basic concepts of optimization and statistical inference which are provided in appendices in sufficient detail. These basic concepts are combined to form the basis of an optimal inference technique for signal detection. We prove a central theorem that characterizes optimality in a variety of distributed detection architectures. We discuss three applications of this result in distributed signal detection. These applications include interactive distributed detection, optimal tandem fusion architecture, and distributed detection by acyclic graph networks. In the conclusion we indicate several future research directions, which include possible generalizations of our optimization method and new research problems arising from each of the three applications considered