Phase transitions in nonlinear filtering

Abstract

It has been established under very general conditions that the ergodic properties of Markov processes are inherited by their conditional distributions given partial information. While the existing theory provides a rather complete picture of classical filtering models, many infinite-dimensional problems are outside its scope. Far from being a technical issue, the infinite-dimensional setting gives rise to surprising phenomena and new questions in filtering theory. The aim of this paper is to discuss some elementary examples, conjectures, and general theory that arise in this setting, and to highlight connections with problems in statistical mechanics and ergodic theory. In particular, we exhibit a simple example of a uniformly ergodic model in which ergodicity of the filter undergoes a phase transition, and we develop some qualitative understanding as to when such phenomena can and cannot occur. We also discuss closely related problems in the setting of conditional Markov random fields.

Can local particle filters beat the curse of dimensionality?

“…While the above back-of-envelope computation provides a basic template for our approach, the rigorous implementation of these ideas requires the introduction of mathematical machinery that has not previously been applied in the study of nonlinear filtering. Just as in the case of the filter stability property (see [18] and the references therein), it is far from clear that any decay of correlations properties of the underlying model are inherited by the filter as we have taken for granted above: in fact, striking counterexamples show that such inheritance can fail in surprising ways [13]. More generally, the development of machinery for the local analysis of high-dimensional filtering problems forms an essential part of our proofs.…”

“…[5]) are illsuited to the investigation of the much more delicate problems that arise in high dimension. We have recently begun to explore high-dimensional probabilistic phenomena in nonlinear filtering [13,18]. The present paper arose from the realization that such phenomena are not only of interest in their own right, but that they can provide mechanisms that enable the development and analysis of particle filtering algorithms in high dimension.…”

Exponential forgetting of smoothing distributions for pairwise Markov models

“…Recently, a significant contribution to the study of smoothing probabilities (with continuous state space) was made by van Handel and his colleagues [39,42,41,40,4,43,37,38,35]. Again, most of the papers deals with HMM's, but in [37,38], also more general PMM's are considered.…”

“…Distinct clusters need not be disjoint and a cluster can consist of a single state. The cluster assumption states: There exists a cluster C ⊂ Y such that the sub-stochastic matrix P C = (p ij ) i,j∈C is primitive, that is P R C has only positive elements for some positive integer R. Thus the cluster assumptions implies that the Markov cain Y is aperiodic but not vice versa -for a counterexample consider a classical example appearing in [1] (Example 4.3.28) as well as in [3,35]. Let Y = {0, 1, 2, 3}, X = {0, 1} and let the Markov chain Y be be defined by Y k = Y k−1 + U k (mod 4), where {U k } is an i.i.d.…”

“…The assumption 2. about the positivity of densities is often made in literature (e.g.. [2,7,8]). In particular, it is the HMM-version of the nondegeneracy-assumption [41,35]. Of course, the above-mentioned articles deal with continuous state space X , where the technique is different.…”