Decision making under uncertainty with Bayesian filters

Abstract

This work is concerned with exploiting Bayesian filters for decision making under uncertainty. The kind of decision making that is formally suitable for problems requiring finding optimal (non-sensing) actions as well as optimal answers/statements. Specifically, the focus will be on filters for spatial point processes which model nature as a population of indistinguishable objects. Previous works have been limited to translating the problem of point estimation into loss functions compatible with object populations. Whereas the present work systematically constructs a number of novel loss functions that give rise to a class of statistical problems beyond point estimation, which have not been appropriately formalized yet. We obtain closedform solutions to those problems (expressions computing optimal statements and corresponding minimized expected values of loss), and implement the solutions with a variety of approximate filters: the classical PHD filter, the Panjer PHD (PPHD) filter, and the Cardinalized PHD (CPHD) filter. We offer practical interpretations of the introduced problems, such as the estimation of risk value attached to an uncertain object population, and demonstrate selected implementations through numerical simulations. Overall, this work extends the variety of problems solvable using information from Bayesian filters, and reduces the amount of avoidable losses in such problems when compared to conventional approaches.