Optimizing statistical decisions by adding noise

This thesis presents an algorithm to find near-optimal "stochastic resonance" (SR) noise to maximize the expected payoff in statistical decision problems subject to a single inequality constraint on the expected cost. The SR effect or noise benefit occurs when the expected cost satisfies the inequality constraint while the expected payoff in the presence of a noise or randomization is larger than in the case without noise. The payoff and cost functions are real-valued bounded nonnegative Borel-measurable functions on a finite-dimensional noise space N . We show that the optimal SR noise is just the randomization of two noise realizations if the statistical decision problem is subject only to a single inequality constraint and if the optimal noise exists. We give necessary and sufficient conditions for the existence of such optimal noise. If the optimal noise does not exist then there exists a sequence of noise random variables such that the limit of the respective expected-payoff sequence is optimal. We develop an algorithm that finds an SR noise N' from a finite set of noise realizations N&d5; ⊆ N . This noise N' is nearly optimal if the payoff function on the actual noise space N is sufficiently close to its restriction to N&d5; . An upper bound limits the number of iterations that the algorithm requires to find such near-optimal SR noise. Two applications demonstrate the SR noise algorithm. The first application finds a near-optimal SR noise for a suboptimal one-sample Neyman-Pearson hypothesis test of variance. The second application gives a near-optimal signal power randomization for an average-power-constrained anti-podal signal transmitter in the presence of additive Gaussian-mixture channel noise where the receiver uses a maximum a posteriori (MAP) method for optimal signal detection. These applications show that the algorithm finds near-optimal noise or randomization in just a few iterations. The algorithm has potential applications in many signal processing and communication systems that have randomized optimal solutions.