Bayesian Estimation with Uncertain Parameters of Probability Density Functions

In this paper, we address the problem of processing imprecisely known probability density functions by means of Bayesian estimation. The imprecise knowledge about probability density functions is given as stochastic uncertainty about their parameters. The proposed processing of this special density in a Bayesian estimator is accomplished by reinterpretation of the filter and prediction equations. Here, the parameters are treated as a higher order state, which can be processed by Bayesian estimation techniques. For state estimation, this avoids the need to select specific values for unknown parameters and, thus, allows the processing of all potential parameters at once. The proposed approach further allows the use of imprecisely known model equations for measurement and state prediction by the same principle.