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Houssineau, Jeremie, Jasra, Ajay and Singh, Sumeetpal S. (2019) On large lag smoothing for hidden Markov Models. SIAM Journal on Numerical Analysis, 57 (6). pp. 2812-2828. doi:10.1137/18M1198004
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Official URL: https://doi.org/10.1137/18M1198004
Abstract
In this article we consider the smoothing problem for hidden Markov models. Given a hidden Markov chain $\{X_n\}_{n\geq 0}$ and observations $\{Y_n\}_{n\geq 0}$, our objective is to compute $\mathbb{E}[\varphi(X_0,\dots,X_k)|y_{0},\dots,y_n]$ for some real-valued, integrable functional $\varphi$ and $k$ fixed, $k \ll n$ and for some realization $(y_0,\dots,y_n)$ of $(Y_0,\dots,Y_n)$. We introduce a novel application of the multilevel Monte Carlo method with a coupling based on the Knothe--Rosenblatt rearrangement. We prove that this method can approximate the aforementioned quantity with a mean square error (MSE) of $\mathcal{O}(\epsilon^2)$ for arbitrary $\epsilon>0$ with a cost of $\mathcal{O}(\epsilon^{-2})$. This is in contrast to the same direct Monte Carlo method, which requires a cost of $\mathcal{O}(n\epsilon^{-2})$ for the same MSE. The approach we suggest is, in general, not possible to implement, so the optimal transport methodology of [A. Spantini, D. Bigoni, and Y. Marzouk, J. Mach. Learn. Res., 19 (2018), pp. 2639--2709; M. Parno, T. Moselhy, and Y. Marzouk, SIAM/ASA J. Uncertain. Quantif., 4 (2016), pp. 1160--1190] is used, which directly approximates our strategy. We show that our theoretical improvements are achieved, even under approximation, in several numerical examples.
| Item Type: | Journal Article | |||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||
| Divisions: | Faculty of Science > Statistics | |||||||||
| Library of Congress Subject Headings (LCSH): | Smoothing (Statistics), Markov processes, Monte Carlo method | |||||||||
| Journal or Publication Title: | SIAM Journal on Numerical Analysis | |||||||||
| Publisher: | Society for Industrial and Applied Mathematics | |||||||||
| ISSN: | 0036-1429 | |||||||||
| Official Date: | 3 December 2019 | |||||||||
| Dates: |
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| Volume: | 57 | |||||||||
| Number: | 6 | |||||||||
| Page Range: | pp. 2812-2828 | |||||||||
| DOI: | 10.1137/18M1198004 | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Publisher Statement: | “First Published in SIAM Journal on Numerical Analysis in 57(6) 2019, published by the Society for Industrial and Applied Mathematics (SIAM)” Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.” | |||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||
| Copyright Holders: | © 2019, Society for Industrial and Applied Mathematics | |||||||||
| RIOXX Funder/Project Grant: |
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