Intrinsically Optimal Bayesian Robust Filtering

When designing optimal filters it is often unrealistic to assume that the statistical model is known perfectly. The issue is then to design a robust filter that is optimal relative to an uncertainty class of processes. Robust filter design has been treated from minimax (best worst-case performance) and Bayesian (best average performance) perspectives. Heretofore, the Bayesian approach has involved finding a model-specific optimal filter, one that is optimal for some model in the uncertainty class. Lifting this constraint, we optimize over the full class from which the original optimal filters were obtained, for instance, over all linear filters. By extending the original characteristics determining the filter, such as the power spectral density, to “effective characteristics” that apply across the uncertainty class, we demonstrate, for both linear and morphological filtering, that an “intrinsically optimal” Bayesian robust filter can be represented in the same form as the standard solution to the optimal filter, except via the effective characteristics. Solutions for intrinsic Bayesian robust filters are more transparent and intuitive than solutions for model-specific filters, and also less tedious, because effective characteristics push through the spectral theory into the Bayesian setting, whereas solutions in the model-specific case depend on grinding out the optimization.