Progressive correction for deterministic Dirac mixture approximations

Since the advent of Monte-Carlo particle filtering, particle representations of densities have become increasingly popular due to their flexibility and implicit adaptive resolution. In this paper, an algorithm for the multiplication of a systematic Dirac mixture (DM) approximation with a continuous likelihood function is presented, which applies a progressive correction scheme, in order to avoid the particle degeneration problem. The preservation of sample regularity and therefore, representation quality of the underlying smooth density, is ensured by including a new measure of smoothness for Dirac mixtures, the DM energy, into the distance measure. A comparison to common correction schemes in Monte-Carlo methods reveals large improvements especially in cases of small overlap between the likelihood and prior density, as well as for multi-modal likelihoods.