On Error Exponents in Hypothesis Testing

The problem of multiple hypothesis testing (HT) for arbitrarily varying sources (AVS) is considered. The achievable error probability exponents (reliabilities) region is derived, optimal decision schemes are described. The result extends the known ones by Fu and Shen and by Tuncel. The Chernoff bounds for AVS binary and M-ary HT are specified via indication of a Sanov theorem for those sources.

The control of the defective rate of spare parts and finished products in the production process of enterprises is very important for product quality and economic benefit. This paper probes into whether the optimal decision scheme is made for the products generated at each stage in the production process after an enterprise gets the defective rate from sampling inspection, and comprehensively uses the multi-stage decision model and simulated annealing algorithm to discuss and answer. Firstly, a decision tree model is established for a simple process, an objective function is set up to maximize profit, and the possible decisions and results of each link are calculated. Then the optimal decision scheme and results are obtained by simulated annealing algorithm. Secondly, for more complex processes, a comprehensive multi-stage decision model is established, the possible decisions and results of each part and each process are calculated, the objective function of total profit is set up, and the optimal scheme is solved by simulated annealing algorithm. The results show that the decision tree model and simulated annealing algorithm can effectively solve the optimization problem of multi-stage detection decision in the production process of enterprises, so as to help enterprises achieve profit maximization.

A central challenge in hypothesis testing (HT) lies in determining the optimal balance between Type I (false positive) and Type II (non-detection or false negative) error probabilities. Analyzing these errors' exponential rate of convergence, known as error exponents, provides crucial insights into system performance. Error exponents offer a lens through which we can understand how operational restrictions, such as resource constraints and impairments in communications, affect the accuracy of distributed inference in networked systems. This survey presents a comprehensive review of key results in HT, from the foundational Stein's Lemma to recent advancements in distributed HT, all unified through the framework of error exponents. We explore asymptotic and non-asymptotic results, highlighting their implications for designing robust and efficient networked systems, such as event detection through lossy wireless sensor monitoring networks, collective perception-based object detection in vehicular environments, and clock synchronization in distributed environments, among others. We show that understanding the role of error exponents provides a valuable tool for optimizing decision-making and improving the reliability of networked systems.

Based on the Joint Video Team (JVT) of the ITU-T Video Coding Experts Group VC EG and the IS O/IEC Moving Picture Experts Group MPEG, an RD optimal Macro Block mode decision scheme for Internet error channel streaming is introduced. The scheme employs the luminance Rate Distortion (RD) optimal mode decision scheme so as to take the effects of video encoding distortion and the channel error propagation to get higher error robustness for error transmission. Based on the Wireless Sensor Network, this paper analyzes the data distortion problem when transmitting H.264 coded video stream over error-prone channel. And the authors also have discussed a widely accepted technique that introduces more intra-coded information on macro block basis. Additionally, this paper introduces a simple loss and multiplication factor estimation method, the rate-distortion optimized assessing strategy over the whole situation.

A distributed binary hypothesis testing (HT) problem involving two parties, one referred to as the observer and the other as the detector is studied. The observer observes a discrete memoryless source (DMS) and communicates its observations to the detector over a discrete memoryless channel (DMC). The detector observes another DMS correlated with that at the observer, and performs a binary hypothesis test on the joint distribution of the two DMS's using its own observed data and the information received from the observer. The trade-off between the type I error probability and the type II error-exponent of the HT is explored. Single-letter lower bounds on the optimal type II error-exponent are obtained by using two different coding schemes, a separate HT and channel coding scheme and a joint HT and channel coding scheme based on hybrid coding for the matched bandwidth case. Exact single-letter characterization of the same is established for the special case of testing against conditional independence, and it is shown to be achieved by the separate HT and channel coding scheme. An example is provided where the joint scheme achieves a strictly better performance than the separation based scheme.

In this paper, we propose a theoretical framework for cooperative abnormality detection and localization systems by exploiting molecular communication setup. The system consists of mobile sensors in a fluidic medium, which are injected into the medium to search the environment for abnormality. Some fusion centers (FC) are placed at specific locations in the medium, which absorb all sensors arrived at their locations, and by observing its state, each FC decides on the abnormality existence and/or its location. To reduce the effects of sensor imperfection, we propose a scheme where the sensors release some molecules (i.e., markers) into the medium after they sense an abnormality. If the goal is abnormality detection, the released molecules are used to cooperatively activate other sensors. If the goal is abnormality localization, the released molecules are used by the FCs to determine the location. In our model, both sensors' imperfection and markers background noise are taken into account. For the detection phase, we consider two sensor types based on their activation strategy by markers. To make the analysis tractable, we assume some ideal assumptions for the sensors' model. We investigate the related binary hypothesis testing problem and obtain the probabilities of false alarm and miss-detection. It is shown that using sensors with the ability of cooperatively activating each other can significantly improve the detection performance in terms of probability of error. For the localization phase, we consider two types of FCs based on their capability in reading sensors' storage levels. We study their performance and obtain the optimal and sub-optimal decision schemes and also the probability of localization error for both perfect and imperfect sensing regimes.

This paper studies binary hypothesis testing with a single sensor that communicates with two decision centers over a memoryless broadcast channel. The main focus lies on the tradeoff between the two type-II error exponents achievable at the two decision centers. In our proposed scheme, we can partially mitigate this tradeoff when the transmitter has a probability larger than 1/2 to distinguish the alternate hypotheses at the decision centers, i.e., the hypotheses under which the decision centers wish to maximize their error exponents. In the cases where these hypotheses cannot be distinguished at the transmitter (because both decision centers have the same alternative hypothesis or because the transmitter’s observations have the same marginal distribution under both hypotheses), our scheme shows an important tradeoff between the two exponents. The results in this paper thus reinforce the previous conclusions drawn for a setup where communication is over a common noiseless link. Compared to such a noiseless scenario, here, however, we observe that even when the transmitter can distinguish the two hypotheses, a small exponent tradeoff can persist, simply because the noise in the channel prevents the transmitter to perfectly describe its guess of the hypothesis to the two decision centers.

Wireless sensor networks are prone to node misbehavior arising from tampering by an adversary (Byzantine attack), or due to other factors such as node failure resulting from hardware or software degradation. In this paper, we consider the problem of decentralized detection in wireless sensor networks in the presence of one or more classes of misbehaving nodes. Binary hypothesis testing is considered where the honest nodes transmit their binary decisions to the fusion center (FC), while the misbehaving nodes transmit fictitious messages. The goal of the FC is to identify the misbehaving nodes and to detect the state of nature. We identify each class of nodes with an operating point (false alarm and detection probabilities) on the receiver operating characteristic (ROC) curve. Maximum likelihood estimation of the nodes' operating points is then formulated and solved using the expectation maximization (EM) algorithm with the nodes' identities as latent variables. The solution from the EM algorithm is then used to classify the nodes and to solve the decentralized hypothesis testing problem. Numerical results compared with those from the reputation-based schemes show a significant improvement in both classification of the nodes and hypothesis testing results. We also discuss an inherent ambiguity in the node classification problem which can be resolved if the honest nodes are in majority.

Achievable error exponents for the one-way with noisy feedback and two-way AWGN channels are derived for the transmission of a finite number of messages $M$ using fixed block length $n$, under the almost sure (AS) and the expected block (EXP) power constraints. In the one-way setting under noisy AWGN feedback, it is shown that under the AS constraint and when the feedback link is much stronger than the direct link, active feedback leads to a larger gain over the non-feedback error exponent than passive feedback. Under the EXP constraint, a previously known error exponent for the transmission of two messages is generalized to any arbitrary but finite number of messages $M$. In the two-way setting, where each user has its own message to send in addition to (possibly) aiding in the transmission of feedback for the opposite direction, error exponent regions are defined and derived for the first time for the AWGN two-way channel under both AS and EXP power constraints. It is shown that feedback or interaction may lead to error exponent gains in one direction, possibly at the expense of a decrease in the error exponents attained in the other direction. The relationship between $M$ and $n$ supported by our achievability strategies is explored.

This work studies error exponent limits in hypothesis testing (HT) in a distributed scenario with partial communication constraints. We derive general conditions on the Type I error restriction under which the error exponent of the optimal Type II error has a closed-form characterization for the task of testing against independence. We show that the error exponent is preserved for a family of decreasing Type I error restrictions. Complementing this analysis, new expressions are derived to bound the optimal Type II error probability for a finite number of observations. These bounds shed light about the velocity at which error exponent limits are attained with the number of samples.

We consider Bayesian multiple hypothesis problem with independent and identically distributed observations. The classical, Sanov's theorem-based, analysis of the error probability allows one to characterize the best achievable error exponent. However, this analysis does not generalize to the case where the true distributions of the hypothesis are not exact or partially known via some nominal distributions. This problem has practical significance, because the nominal distributions may be quantized versions of the true distributions in a hardware implementation, or they may be estimates of the true distributions obtained from labeled training sequences as in statistical classification. In this paper, we develop a type-based analysis to investigate Bayesian multiple hypothesis testing problem. Our analysis allows one to explicitly calculate the error exponent of a given type and extends the classical analysis. As a generalization of the proposed method, we derive a robust test and obtain its error exponent for the case where the hypothesis distributions are not known but there exist nominal distribution that are close to true distributions in variational distance.

Suppose Q is a family of discrete memoryless channels. An unknown member of Q will be available, with perfect, causal output feedback for communication. Is there a coding scheme (possibly with variable transmission time) that can achieve the Burnashev error exponent uniformly over Q ? For two families of channels we show that the answer is yes. Furthermore, for each of these two classes, in addition to achieve the maximum error exponent, it is possible to uniformly attain any given fraction of the channel capacity. Therefore, in terms of achievable rates and delay, there are situations in which the knowledge of the channel becomes irrelevant. In the second part of the thesis, we show that for arbitrary sets of channels the Burnashev error exponent cannot in general be uniformly achieved. In particular we give a sufficient condition for a pair of channels so that no coding strategy reaches Burnashev's exponent simultaneously on both channels. As a third part we study a scenario where communication is carried by first testing the channel by means of a training sequence, then coding according to the channel estimate. We provide an upper bound on the maximum achievable error exponent of such coding schemes. This bound is typically much lower than the maximum achievable error exponent over a channel with feedback. For example in the case of binary symmetric channels this bound has a slope that vanishes at capacity. This result suggests that in terms of error exponent, a good universal feedback scheme combines channel estimation with information delivery, rather than separating them. In the final chapter, we address the question of communicating quickly and reliably. We consider a simple situation of two message communication over a known channel with feedback. We propose a simple decoding rule, and show that it minimizes a weighted combination of the probability of error and decoding delay for a certain range of crossover probabilities and combination weights.

Fundamental neural network structures in decentralized hypothesis testing are considered. For binary hypothesis testing, the basic neural operations are established, and the Neyman-Pearson criterion is utilized due to information theoretic arguments. Then, two fundamental neural structures are considered, and analyzed and compared in terms of asymptotic performance measures. In particular, the asymptotic relative efficiency performance measure is used to establish performance characteristics and tradeoffs in the two structures, for both parametrically and nonparametrically defined hypotheses. In the latter case, robust neural network structures are considered, and their superiority to parametric network structures is argued. >

In direct sequence code division multiple access (DS-CDMA) systems, signal acquisition is necessary before communication can commence. Recent work has shown that the problem of acquisition may be even more restrictive than the problem of error control in limiting the capacity and performance of these systems. Passive matched filters and parallel search schemes have been shown to be able to acquire signals rapidly, but they do so at the cost of high or prohibitive complexity. In contrast, straight serial search schemes have lower complexity but acquire the signal much more slowly. In this paper, we studyhybrid active correlation schemes that provide flexibility in the trade-off between acquisition speed and complexity. These techniques test several phases concurrently and either decide that a particular phase is correct, in which case the decision is verified by a binary hypothesis test, or that none of the phases are correct, in which case another group of phases is tested. Two new hybrid acquisition schemes are presented; the first is based on a fixed-sample-size weighted MAP test, and the second is based on anM-ary sequential hypothesis test called MSPRT. It is shown that the weighted MAP-based scheme hypothesis test called MSPRT. It is shown that the weighted MAP-based scheme outperforms a standard MAP-based scheme. It is also shown that considerable performance gain can be obtained using sequential testing. In particular, it is shown through numerical examples that MSPRT-based schemes are several times faster than corresponding fixed-sample-size schemes.

Binary hypothesis testing using structured adaptive networks