Distributional Kalman filters for Bayesian forecasting and closed form recurrences

Abstract

Over the last 50 years there has been an enormous explosion in developing full distributional analogues of the Kalman filter. In this paper we explore how some of the second-order processes discovered by Kalman have their analogues in Bayesian state space models. Many of the analogues in the lierature need to be calculated using numerical methods like Markov chain Monte Carlo so they retain, or even enhance, the descriptive power of the Kalman filter, but at the cost of reduced transparency. However, if the analogues are drawn properly, elegant recurrence relationshipsa-like those of the Kalman filtera-can still be developed that apply, at least, for one-step-ahead forecast distributions. In this paper we explore the variety of ways such models have been built, in particular with respect to graphical time series models.