Bayesian estimation with imprecise likelihoods in the framework of random set theory

The problem is Bayesian estimation in the circumstances where the the likelihood functions are only partially known. The measurement model in this case is affected by two sources of uncertainty: the stochastic uncertainty and imprecision. Following the framework of random set theory, the paper presents the optimal Bayesian estimator for this problem. The resulting Bayes estimator in general has no analytic closed form solution, but can be approximated, for example, using the Monte Carlo method. The proposed Bayesian estimator is illustrated by a source localisation example, where imprecision was due to the partial knowledge of sensor locations and propagation loss factors. It has been verified by Monte Carlo simulations that the support of an accurate approximation of the posterior PDF is guaranteed to contain the true value of the parameter vector.