Better Bounds for Bayesian Multiple Test with Quadratic Loss Function

A Bayesian test has been previously proposed for a multiple hypothesis testing problem given the 0-1 loss function. However, this function is not suitable for many applications such as intrusion detection, anomaly detection where a quadratic loss function can be more appropriate to distinguish the concurrent hypotheses. Although a Bayesian test with the quadratic loss function has been constructed for this problem, its asymptotic performance has not yet been well studied due to its poor bounds. The main contribution of this paper is the construction of better bounds for this Bayesian test and the one associated with the 0-1 loss function. With these new bounds, it is theoretically established that the asymptotic equivalence between these two tests depends on the geometry of the parameter space associated with the hypotheses.