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Stable approximation schemes for optimal filters

2 September 2018
Dan Crisan
A. López-Yela
Joaquín Míguez
ArXiv (abs)PDFHTML
Abstract

We explore a general truncation scheme for the approximation of (possibly unstable) optimal filters. In particular, let \mS=(π0,κt,gt)\mS = (\pi_0,\kappa_t,g_t)\mS=(π0​,κt​,gt​) be a state space model defined by a prior distribution π0\pi_0π0​, Markov kernels {κt}t≥1\{\kappa_t\}_{t\ge 1}{κt​}t≥1​ and potential functions {gt}t≥1\{g_t\}_{t \ge 1}{gt​}t≥1​, and let \sfc={Ct}t≥1\sfc = \{C_t\}_{t\ge 1}\sfc={Ct​}t≥1​ be a sequence of compact subsets of the state space. In the first part of the manuscript, we describe a systematic procedure to construct a system \mS\sfc=(π0,κt\sfc,gt\sfc)\mS^\sfc=(\pi_0,\kappa_t^\sfc,g_t^\sfc)\mS\sfc=(π0​,κt\sfc​,gt\sfc​), where each potential gt\sfcg_t^\sfcgt\sfc​ is truncated to have null value outside the set CtC_tCt​, such that the optimal filters generated by SSS and S\sfcS^\sfcS\sfc can be made arbitrarily close, with approximation errors independent of time ttt. Then, in a second part, we investigate the stability of the approximately-optimal filters. Specifically, given a system \mS\mS\mS with a prescribed prior π0\pi_0π0​, we seek sufficient conditions to guarantee that the truncated system \mS\sfc\mS^\sfc\mS\sfc (with {\em the same} prior π0\pi_0π0​) generates a sequence of optimal filters which are stable and, at the same time, can attain arbitrarily small approximation errors. Besides the design of approximate filters, the methods and results obtained in this paper can be applied to determine whether a prescribed system \mS\mS\mS yields a sequence of stable filters and to investigate topological properties of classes of optimal filters. As an example of the latter, we explicitly construct a metric space (\mfS,Dq)(\mfS,D_q)(\mfS,Dq​), where \mfS\mfS\mfS is a class of state space systems and DqD_qDq​ is a proper metric on \mfS\mfS\mfS, which contains a dense subset \mfS0⊂\mfS\mfS_0 \subset \mfS\mfS0​⊂\mfS such that every element \mS0∈\mfS0\mS_0 \in \mfS_0\mS0​∈\mfS0​ is a state space system yielding a stable sequence of optimal filters.

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