Noise Enhanced Hypothesis-Testing in the Restricted Bayesian Framework

Abstract

Abstract-Performance of some suboptimal detectors can be enhanced by adding independent noise to their observations. In this paper, the effects of additive noise are investigated according to the restricted Bayes criterion, which provides a generalization of the Bayes and minimax criteria. Based on a generic -ary composite hypothesis-testing formulation, the optimal probability distribution of additive noise is investigated. Also, sufficient conditions under which the performance of a detector can or cannot be improved via additive noise are derived. In addition, simple hypothesis-testing problems are studied in more detail, and additional improvability conditions that are specific to simple hypotheses are obtained. Furthermore, the optimal probability distribution of the additive noise is shown to include at most mass points in a simple -ary hypothesis-testing problem under certain conditions. Then, global optimization, analytical and convex relaxation approaches are considered to obtain the optimal noise distribution. Finally, detection examples are presented to investigate the theoretical results.

Noise benefits in joint detection and estimation problems

“…Using the primal-dual concept, [23] reaches PMFs with at most two point masses under certain conditions for binary hypothesis testing problems. In [29] and [25], the proof given in [15] is extended to hypothesis testing problems with ðM À1Þ constraint functions and the optimum noise distribution is found to have M point masses.…”

“…The structures of the proposed problems are similar to those in [15,23,25,29]. The same principles can be applied to both of the optimization problems in (12) and (21) and the optimum noise distribution structure can be specified under certain conditions as follows:…”

“…equivalently, V [15,23,25,29]. Then, from Carathéodory's theorem [40], it can be concluded that any point on the boundary of V can be expressed as the convex combination of at most three different points in U.…”

“…They examined the convex structure of the problem and specified the nature of the optimal probability distribution of additive noise as a probability mass function with at most two point masses. This result is generalized for M-ary composite hypothesis testing problems under NP, restricted NP and restricted Bayes criteria [25,29,34]. In estimation problems, additive noise can also be utilized to improve the performance of a given suboptimal estimator [4,19,30].…”

“…Later, it is proved that a suboptimal detector in the Bayesian framework may be improved (i.e., the Bayes risk can be reduced) by adding a constant signal to the observation signal; that is, the optimal probability density function is a single Dirac delta function [14]. This intuition is extended in various directions and it is demonstrated that injection of additive noise to the observation signal at the input of a suboptimal detector can enhance the system performance [15,[18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. In this paper, performance improvements through noise benefits are addressed in the context of joint detection and estimation systems by adding an independent noise component to the observation signal at the input of a suboptimal system.…”