Smoothing With Couplings of Conditional Particle Filters

Jacob, Pierre; Lindsten, Fredrik; Schön, Thomas B.

doi:10.1080/01621459.2018.1548856

Cited by 43 publications

(66 citation statements)

References 68 publications

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“…Explicit constructions of coupled chains satisfying Assumptions 1-3 for Markov kernels K that are defined by Metropolis-Hastings algorithms and Gibbs samplers are given in Jacob et al [2017, Section 4] and Jacob et al [2018]. The focus of this article is to propose a coupling strategy that is tailored for Hamiltonian Monte Carlo chains, so as to enable the use of unbiased estimators (1)-(2).…”

Section: Unbiased Estimation With Couplingsmentioning

confidence: 99%

Unbiased Hamiltonian Monte Carlo with couplings

Heng¹,

Jacob

2019

Biometrika

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We propose a methodology to parallelize Hamiltonian Monte Carlo estimators. Our approach constructs a pair of Hamiltonian Monte Carlo chains that are coupled in such a way that they meet exactly after some random number of iterations. These chains can then be combined so that resulting estimators are unbiased. This allows us to produce independent replicates in parallel and average them to obtain estimators that are consistent in the limit of the number of replicates, instead of the usual limit of the number of Markov chain iterations. We investigate the scalability of our coupling in high dimensions on a toy example. The choice of algorithmic parameters and the efficiency of our proposed methodology are then illustrated on a logistic regression with 300 covariates, and a log-Gaussian Cox point processes model with low to fine grained discretizations.

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Section: Unbiased Estimation With Couplingsmentioning

confidence: 99%

Unbiased Hamiltonian Monte Carlo with couplings

Heng¹,

Jacob

2019

Biometrika

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“…The CPF developed in [6] (see also [16], [17], [22], [23], and [28]) is used in several applications as discussed above and various other contexts. It consists of a particle filter which runs on the product space of the two filters.…”

Section: Introductionmentioning

confidence: 99%

Central limit theorems for coupled particle filters

Jasra

2020

Adv. Appl. Probab.

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In this article we prove new central limit theorems (CLTs) for several coupled particle filters (CPFs). CPFs are used for the sequential estimation of the difference of expectations with respect to filters which are in some sense close. Examples include the estimation of the filtering distribution associated to different parameters (finite difference estimation) and filters associated to partially observed discretized diffusion processes (PODDP) and the implementation of the multilevel Monte Carlo (MLMC) identity. We develop new theory for CPFs, and based upon several results, we propose a new CPF which approximates the maximal coupling (MCPF) of a pair of predictor distributions. In the context of ML estimation associated to PODDP with time-discretization $\Delta_l=2^{-l}$ , $l\in\{0,1,\dots\}$ , we show that the MCPF and the approach of Jasra, Ballesio, et al. (2018) have, under certain assumptions, an asymptotic variance that is bounded above by an expression that is of (almost) the order of $\Delta_l$ ( $\mathcal{O}(\Delta_l)$ ), uniformly in time. The $\mathcal{O}(\Delta_l)$ bound preserves the so-called forward rate of the diffusion in some scenarios, which is not the case for the CPF in Jasra et al. (2017).

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“…Firstly, to extend our theoretical results to the case of the approximated coupling. Secondly, to investigate whether the coupling used in [9] can also yield, theoretically, the same improvements that have been seen in the work in this article.…”

Section: Discussionmentioning

confidence: 72%

“…[3,11]) to couple smoothers, versus the optimal Wasserstein coupling. The goal in [9] is unbiased estimation which is complementary to ideas in this article, where we focus upon reducing the cost of large lag smoothing.…”

Section: Case X ⊂ R Dmentioning

confidence: 99%

On Large Lag Smoothing for Hidden Markov Models

Houssineau¹,

Jasra²,

Singh³

2019

SIAM J. Numer. Anal.

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In this article we consider the smoothing problem for hidden Markov models (HMM).Given a hidden Markov chain {Xn} n≥0 and observations {Yn} n≥0 , our objective is to compute E[ϕ(X0, . . . , X k )|y0, . . . , yn] for some real-valued, integrable functional ϕ and k fixed, k n and for some realisation (y0, . . . , yn) of (Y0, . . . , Yn). 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( 2 ), for arbitrary > 0 with a cost of O( −2 ). This is in contrast to the same direct Monte Carlo method, which requires a cost of O(n −2 ) for the same MSE. The approach we suggest is, in general, not possible to implement, so the optimal transport methodology of[17] 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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Smoothing With Couplings of Conditional Particle Filters

Cited by 43 publications

References 68 publications

Unbiased Hamiltonian Monte Carlo with couplings

Unbiased Hamiltonian Monte Carlo with couplings

Central limit theorems for coupled particle filters

On Large Lag Smoothing for Hidden Markov Models

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