On large lag smoothing for hidden Markov models

Type
Article

Authors
Houssineau, Jeremie
Jasra, Ajay
Singh, Sumeetpal S.

KAUST Department
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Online Publication Date
2019-12-03

Print Publication Date
2019-01

Date
2019-12-03

Abstract
In this article we consider the smoothing problem for hidden Markov models. Given a hidden Markov chain { Xn} n≥ 0 and observations { Yn} n≥ 0, our objective is to compute E[varphi (X0, . ,Xk)| y0, . , yn] for some real-valued, integrable functional varphi and k fixed, k ll n and for some realization (y0, . , yn) of (Y0, . , Yn). 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 scrO (∈-2) for arbitrary ∈ > 0 with a cost of scrO (∈-2). This is in contrast to the same direct Monte Carlo method, which requires a cost of scrO (n∈-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.

Citation
Houssineau, J., Jasra, A., & Singh, S. S. (2019). On Large Lag Smoothing for Hidden Markov Models. SIAM Journal on Numerical Analysis, 57(6), 2812–2828. doi:10.1137/18m1198004

Publisher
Society for Industrial & Applied Mathematics (SIAM)

Journal
SIAM Journal on Numerical Analysis

DOI
10.1137/18M1198004

arXiv
1804.07117

Additional Links
https://epubs.siam.org/doi/10.1137/18M1198004http://arxiv.org/pdf/1804.07117

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