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Convergence of Sequential Monte Carlo Methods for Bayesian Inference

This document explores the convergence of Sequential Monte Carlo (SMC) methods in Bayesian inference, illustrating the relationship between prediction and observation processes. It establishes a framework where the signal follows specific evolution equations, and observations adhere to Bayes' recursion. Key steps involve sequential importance sampling, empirical measure formulation, and Markov Chain Monte Carlo (MCMC) techniques for optimizing particle selection based on importance weights. The findings highlight the mean square error convergence under both general and restrictive conditions, aiming to enhance SMC method robustness.

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Convergence of Sequential Monte Carlo Methods for Bayesian Inference

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  1. Convergence of Sequential Monte Carlo Methods Dan Crisan, Arnaud Doucet

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

  3. Bayes’ recursion • Prediction • Updating

  4. 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

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

  6. Step 2: Selection 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

  7. 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

  8. 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

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

  10. 2.-A Resampling/Selection scheme

  11. 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”.

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

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

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

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