TY - JOUR
T1 - Phase transitions in nonlinear filtering
AU - Rebeschini, Patrick
AU - van Handel, Ramon
N1 - Publisher Copyright:
© 2015, University of Washington. All rights reserved.
Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.
PY - 2015
Y1 - 2015
N2 - 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.
AB - 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.
KW - Conditional ergodicity and mixing
KW - Filtering in infinite dimension
KW - Phase transitions
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U2 - 10.1214/EJP.v20-3281
DO - 10.1214/EJP.v20-3281
M3 - Article
AN - SCOPUS:84961376032
VL - 20
JO - Electronic Journal of Probability
JF - Electronic Journal of Probability
SN - 1083-6489
ER -