On Large Lag Smoothing for Hidden Markov Models
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In this article we consider the smoothing problem for hidden Markov models (HMM). Given a hidden Markov chain {X_n}_n≥ 0 and observations {Y_n}_n≥ 0, our objective is to compute E[φ(X_0,...,X_k)|y_0,...,y_n] for some real-valued, integrable functional φ and k fixed, k ≪ n and for some realisation (y_0,...,y_n) of (Y_0,...,Y_n). We introduce a novel application of the multilevel Monte Carlo (MLMC) method with a coupling based on the Knothe-Rosenblatt rearrangement. We prove that this method can approximate the afore-mentioned quantity with a mean square error (MSE) of O(ϵ^2), for arbitrary ϵ>0 with a cost of O(ϵ^-2). This is in contrast to the same direct Monte Carlo method, which requires a cost of O(nϵ^-2) for the same MSE. The approach we suggest is, in general, not possible to implement, so the optimal transport methodology of span is used, which directly approximates our strategy. We show that our theoretical improvements are achieved, even under approximation, in several numerical examples.
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