dc.contributor.author Houssineau, J en dc.contributor.author Jasra, A en dc.contributor.author Singh, Sumeetpal en dc.date.accessioned 2019-09-04T23:30:45Z dc.date.available 2019-09-04T23:30:45Z dc.date.issued 2019-01-01 en dc.identifier.issn 0036-1429 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/296419 dc.description.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 $\mathbb{E}[\varphi(X_0,\dots,X_k)|y_{0},\dots,y_n]$ for some real-valued, integrable functional $\varphi$ and $k$ fixed, $k \ll n$ and for some 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 $\mathcal{O}(\epsilon^2)$, 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 \cite{span, parno} is used, which directly approximates our strategy. We show that our theoretical improvements are achieved, even under approximation, in several numerical examples. dc.publisher Society for Industrial and Applied Mathematics dc.rights All rights reserved dc.rights.uri dc.title On large lag smoothing for hidden markov models en dc.type Article prism.endingPage 2828 prism.issueIdentifier 6 en prism.publicationDate 2019 en prism.publicationName SIAM Journal on Numerical Analysis en prism.startingPage 2812 prism.volume 57 en dc.identifier.doi 10.17863/CAM.43469 dcterms.dateAccepted 2019-08-30 en rioxxterms.versionofrecord 10.1137/18M1198004 en rioxxterms.version AM rioxxterms.licenseref.uri http://www.rioxx.net/licenses/all-rights-reserved en rioxxterms.licenseref.startdate 2019-01-01 en dc.identifier.eissn 1095-7170 rioxxterms.type Journal Article/Review en pubs.funder-project-id Alan Turing Institute (unknown) cam.orpheus.success Thu Jan 30 10:40:58 GMT 2020 - Embargo updated * rioxxterms.freetoread.startdate 2019-01-01
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