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arXiv · 1804.07117

On Large Lag Smoothing for Hidden Markov Models

Jeremie Houssineau·Ajay Jasra·Sumeetpal S. Singh
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Abstract

In this article we consider the smoothing problem for hidden Markov models (HMM). Given a hidden Markov chain $\{X_n\}_{n\geq 0}$ and observations $\{Y_n\}_{n\geq 0}$, our objective is to compute $\mathbb{E}[φ(X_0,\dots,X_k)|y_{0},\dots,y_n]$ for some real-valued, integrable functional $φ$ and $k$ fixed, $k \ll n$ and for some realisation $(y_0,\dots,y_n)$ of $(Y_0,\dots,Y_n)$. 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 $\mathcal{O}(ε^2)$, for arbitrary $ε>0$ with a cost of $\mathcal{O}(ε^{-2})$. This is in contrast to the same direct Monte Carlo method, which requires a cost of $\mathcal{O}(nε^{-2})$ for the same MSE. The approach we suggest is, in general, not possible to implement, so the optimal transport methodology of \cite{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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Jeremie Houssineau, Ajay Jasra, Sumeetpal S. Singh. 2018-04-19. On Large Lag Smoothing for Hidden Markov Models. https://arxiv.org/abs/1804.07117

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