Bayesian Multiple Hypotheses Testing with Quadratic Criterion

The anomaly detection and localization problem can be treated as a multiple hypotheses testing (MHT) problem in the Bayesian framework. The Bayesian test with the 0-1 loss function is a standard solution for this problem, but the alternative hypotheses have quite different importance in practice. The 0-1 loss function does not reflect this fact while the quadratic loss function is more appropriate. The objective of the thesis is the design of a Bayesian test with the quadratic loss function and its asymptotic study. The construction of the test is made in two steps. In the first step, a Bayesian test with the quadratic loss function for the MHT problem without the null hypothesis is designed and the lower and upper bounds of the misclassification probabilities are calculated. The second step constructs a Bayesian test for the MHT problem with the null hypothesis. The lower and upper bounds of the false alarm probabilities, the missed detection probabilities as well as the misclassification probabilities are calculated. From these bounds, the asymptotic equivalence between the proposed test and the standard one with the 0-1 loss function is studied. A lot of simulation and an acoustic experiment have illustrated the effectiveness of the new statistical test.