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Register a new userReplica Conditional Sequential Monte Carlo
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Added by
Alexander Y. Shestopaloff
Publication date
2019
fields
Mathematical Statistics
and research's language is
English
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We propose a Markov chain Monte Carlo (MCMC) scheme to perform state inference in non-linear non-Gaussian state-space models. Current state-of-the-art methods to address this problem rely on particle MCMC techniques and its variants, such as the iterated conditional Sequential Monte Carlo (cSMC) scheme, which uses a Sequential Monte Carlo (SMC) type proposal within MCMC. A deficiency of standard SMC proposals is that they only use observations up to time $t$ to propose states at time $t$ when an entire observation sequence is available. More sophisticated SMC based on lookahead techniques could be used but they can be difficult to put in practice. We propose here replica cSMC where we build SMC proposals for one replica using information from the entire observation sequence by conditioning on the states of the other replicas. This approach is easily parallelizable and we demonstrate its excellent empirical performance when compared to the standard iterated cSMC scheme at fixed computational complexity.
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The iterated conditional sequential Monte Carlo (i-CSMC) algorithm from Andrieu, Doucet and Holenstein (2010) is an MCMC approach for efficiently sampling from the joint posterior distribution of the $T$ latent states in challenging time-series models, e.g. in non-linear or non-Gaussian state-space models. It is also the main ingredient in particle Gibbs samplers which infer unknown model parameters alongside the latent states. In this work, we first prove that the i-CSMC algorithm suffers from a curse of dimension in the dimension of the states, $D$: it breaks down unless the number of samples (particles), $N$, proposed by the algorithm grows exponentially with $D$. Then, we present a novel local version of the algorithm which proposes particles using Gaussian random-walk moves that are suitably scaled with $D$. We prove that this iterated random-walk conditional sequential Monte Carlo (i-RW-CSMC) algorithm avoids the curse of dimension: for arbitrary $N$, its acceptance rates and expected squared jumping distance converge to non-trivial limits as $D to infty$. If $T = N = 1$, our proposed algorithm reduces to a Metropolis--Hastings or Barkers algorithm with Gaussian random-walk moves and we recover the well known scaling limits for such algorithms.
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