{
  "id": "2108.10277",
  "version": "v1",
  "published": "2021-08-23T16:39:05.000Z",
  "updated": "2021-08-23T16:39:05.000Z",
  "title": "Conditional sequential Monte Carlo in high dimensions",
  "authors": [
    "Axel Finke",
    "Alexandre H. Thiery"
  ],
  "comment": "47 pages, 5 figures",
  "categories": [
    "stat.CO"
  ],
  "abstract": "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 Barker's algorithm with Gaussian random-walk moves and we recover the well known scaling limits for such algorithms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-23T16:39:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "high dimensions",
      "gaussian random-walk moves",
      "squared jumping distance converge",
      "random-walk conditional sequential monte carlo",
      "infer unknown model parameters alongside"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}