Convergence of Sequential Monte Carlo Methods Dan Crisan, Arnaud Doucet

Problem Statement X: signal, Y: observation process X satisfies and evolves according to the following equation, Y satisfies

Bayes’ recursion Prediction Updating

A Sequential Monte Carlo Methods Empirical measure Transition kernel Importance distribution : abs. continuous with respect to : strictly positive Radon Nykodym derivative Then is also continuous w.r.t. and

Algorithm Step 1:Sequential importance sampling sample: evaluate normalized importance weights and let

Step 2: Selection step Step 3: MCMC step multiply/discard particles with high/low importance weights to obtain N particles let assoc.empirical measure Step 3: MCMC step sample ,where K is a Markov kernel of invariant distribution and let

Convergence Study denote convergence to 0 of average mean square error under quite general conditions Then prove (almost sure) convergence of toward under more restrictive conditions

Bounds for mean square errors Assumptions 1.-A Importance distribution and weights is assumed abs.continuous with respect to for all is a bounded function in argument define

There exists a constant s. t. for all there exists with s.t. There exists s. t. and a constant s.t.

2.-A Resampling/Selection scheme

First Assumption ensures that Importance function is chosen so that the corresponding importance weights are bounded above. Sampling kernel and importance weights depend “ continuously” on the measure variable. Second assumption ensures that Selection scheme does not introduce too strong a “discrepancy”.

Lemma 1 Lemma 2 Let us assume that for any then after step 1, for any then for any

Lemma 3 Lemma 4 Let us assume that for any then after step 2, for any then for any

Theorem 1 For all , there exists independent of s.t. for any