In this article these two-filter results and some new ones are derived in a simple way in a very general setting (for arbitrary nonstationary processes). It turns out however that only if a wide-sense (i.e. second order) Markovian assumption is added can one of the filters be viewed as a backwards filter. The remainder of the paper is organized as follows. Section 2 recalls some smoothing formulae that apply to both continuous and discrete observations. Section 3 discusses two types of two-filter-like formulae for general nonstationary processes. In Section 4 one of the filters is shown to be a backwards least squares estimate provided a wide sense Markovian assumption is satisfied. Section 5 contains a derivation of some backwards filters. In Section 6 some additional two-filter-like formulae are given. The final section is a conlusion.

Recently a great amount of attention has been focused on various algorithms for solving the smoothing problem of linear estimation theory. This work is the first part of a two part investigation of these algorithms. In Part I it is shown how change of initial condition (CIC) or partitioning formulae hold in a very general setting (the CIC problem is shown to involve fixed rank perturbation in matrix inversion). In Part II the nature of the two-filter algorithms is explored by providing a simple derivation that shows to what extent the formulae hold generally and so reveals exactly how a wide sense Markovian assumption is necessary for their full utility. The remainder of the paper is structured as follows. Section I contains a discussion of CIC formulae for discrete observations. Section II concerns CIC formulae for continuous observations (actually the formulae are the same). Section III discusses the relation with other work.

A unified Bayesian treatment of the state-of-the-art filtering, smoothing, and parameter estimation algorithms for non-linear state space models.