A minimax approach to Bayesian estimation with partial knowledge of the observation model

We address the problem of Bayesian estimation where the statistical relation between the signal and measurements is only partially known. We propose modeling partial Baysian knowledge by using an auxiliary random vector called instrument. The joint probability distributions of the instrument and the signal, and of the instrument and the measurements, are known. However, the joint probability function of the signal and measurements is unknown. Our model generalizes that underlying the method of instrumental variables from statistics, in that the instrument does not have to satisfy any requirements and no parametric form for the optimal regressor needs to be available. We begin by deriving an estimator for this scenario, via a worst-case design strategy. We then propose a non-parametric method for learning this estimator from a set of examples. Finally, we demonstrate our approach in the context of enhancement of facial images that have undergone an unknown degradation.