Recursive Monte Carlo filters: Algorithms and theoretical analysis

The Annals of Statistics · 2005 · Volume 33 · Issue 5

Abstract

Recursive Monte Carlo filters, also called particle filters, are a powerful tool to perform computations in general state space models. We discuss and compare the accept–reject version with the more common sampling importance resampling version of the algorithm. In particular, we show how auxiliary variable methods and stratification can be used in the accept–reject version, and we compare different resampling techniques. In a second part, we show laws of large numbers and a central limit theorem for these Monte Carlo filters by simple induction arguments that need only weak conditions. We also show that, under stronger conditions, the required sample size is independent of the length of the observed series.

Keywords

60G35, 60J22, 62M09, 65C05, auxiliary variables, central limit theorem, filtering and smoothing, Hidden Markov models, Particle filters, sampling importance resampling, state space models

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