## SMC²

Pierre Jacob, along with Nicolas Chopin and Omiros Papaspiliopoulos, completed a massive work on SMC², a sequential Monte Carlo method that builds on the particle MCMC discussion paper of Andrieu, Doucet and Hollenstein. The major advance is to incorporate sequential requirements within the pMCMC scheme for processing state space models,

$y_t|x_{1:t},y_{1:(t-1)},\theta \sim f(y_t|x_t,\theta) \quad x_t|x_{1:(t-1)}\sim q(x_t|x_{t-1},\theta)$

while still estimating the parameters of the model. A preliminary version was put on arXiv in January, but they reworked their examples towards more realistic and complex cases. One of the new examples is an exciting problem about women’s records on 3000m with an incredible gain of 16 seconds in 1993.

Pierre gave us a presentation at CREST last morning, whose slides are available on slideshare. (The beginning is a standard reminder on sequential Monte Carlo in state-space/hidden Markov models. A question that popped out during the talk is whether or not the estimator of the marginal likelihood derived from the weights of the sequential methods was trustworthy. The error rate is in O(T).) I recall that pMCMC (discussed in this post) is a regular (valid) MCMC algorithm that involves a SMC at each iteration leading to use the approximation of the marginal likelihood $p(y_{1:T}|\theta)$ within the Metropolis-Hastings acceptance ratio. SMC² stands for “SMC on SMC” and it indeed uses a sequential particle approximation to run the pMCMC on a sequential basis. The paper establishes the validity of the scheme via a (huge) completion argument. The examples provided therein are quite complex hidden Markov models and the performances of the SMC² fairly convincing.

### One Response to “SMC²”

1. […] Acrobat… (Most items are working, though, including the bluetooth Apple mouse… Anyway, Pierre kindly offered to take a look over the weekend.) This was not the end of my worries, though, as I […]

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