Conditional sequential Monte Carlo in high dimensions

Abstract

The iterated conditional sequential Monte Carlo (i-CSMC) algorithm from Andrieu, Doucet and Holenstein (J. R. Stat. Soc. Ser. B Stat. Methodol. 72 (2010) 269–342) is an MCMC approach for efficiently sampling from the joint posterior distribution of the T latent states in challenging time-series models, for example, in nonlinear 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 nontrivial limits as $\mathit{D}\to \infty$. If $\mathit{T}=\mathit{N}=1$, our proposed algorithm reduces to a Metropolis–Hastings or Barker’s algorithm with Gaussian random-walk moves and we recover the well-known scaling limits for such algorithms.