Stable approximation schemes for optimal filters
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We explore a general truncation scheme for the approximation of (possibly unstable) optimal filters. In particular, let = (π_0,κ_t,g_t) be a state space model defined by a prior distribution π_0, Markov kernels {κ_t}_t> 1 and potential functions {g_t}_t > 1, and let = {C_t}_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 ^=(π_0,κ_t^,g_t^), where each potential g_t^ is truncated to have null value outside the set C_t, such that the optimal filters generated by S and S^ can be made arbitrarily close, with approximation errors independent of time t. Then, in a second part, we investigate the stability of the approximately-optimal filters. Specifically, given a system with a prescribed prior π_0, we seek sufficient conditions to guarantee that the truncated system ^ (with the same prior π_0) 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 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 (,D_q), where is a class of state space systems and D_q is a proper metric on , which contains a dense subset _0 ⊂ such that every element _0 ∈_0 is a state space system yielding a stable sequence of optimal filters.
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