Approximate Non-Gaussian Bayesian Estimation and Modal Consistency

SUMMARY A new recursive estimation procedure is proposed for the location of a dynamic linear model with non-normal errors. The procedure is a modification of a modal approximation algorithm, which is shown to be prone to instabilities. The modification is motivated by a notion of posterior modal consistency. Many researchers have considered the problem of sequential updating of the first two posterior moments of the location vector of a dynamic linear time series model with non-normal errors. Such models are motivated by considerations of realism or robustness (in particular accommodation of outliers). In this paper, we shall re- examine, for the scalar case, a posterior modal approximation scheme discussed by West (1981) and Fahrmeir and Kaufmann (1991), which in practice has been found to be prone to producing wild instabilities in the recursive estimates. We identify the cause of this and present a modified form of recursive approximation which avoids this problem. We assume fully specified measurement and system models, a standard assumption for the autonomous tracking filters that we have in mind as applications. An extension to unknown measurement variance, along the lines of West (1981), is straightforward.