Optimal Joint Detection and Estimation Based on Decision-Dependent Bayesian Cost

Abstract

This paper considers the statistical inference problem where both the hypothesis testing and the parameter estimation are of primary interest. With unknown parameters present in each hypothesis, the goal is to detect the true hypothesis and to estimate the unknown parameters simultaneously. In light of the coupling nature of these two subproblems, we adopt the Bayesian estimation cost function that depends on both the detection result and the estimation scheme. We then obtain the optimal joint detector and estimator by minimizing the Bayesian estimation cost subject to the constraint on detection performance. The proposed joint solution not only yields lower estimation cost compared with the method that treats the detection and estimation separately, but also allows for flexible tradeoff between the performances of these two subproblems. In addition, we also extend our framework to the multi-hypothesis scenario, where $K$ hypotheses and their associated unknown parameters are present. Finally, we apply the proposed joint detection and estimation framework to the spectrum sensing for cognitive radio, which aims to detect the presence of the primary user and to estimate the noise/interference level at the same time.