Conditional sequential Monte Carlo in high dimensions

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 statespace 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→∞. If T = 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.