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Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. (English) Zbl 1079.65006

A particle system is a collection ${\left({\vartheta }^{\left(j,H\right)},{w}^{\left(j,H\right)}\right)}_{j\le H}$ where ${\vartheta }^{\left(j,H\right)}\in {\Theta }$ are “particles” and ${w}^{\left(j,H\right)}>0$ are their “weights”. The system targets a distribution $\pi$ on ${\Theta }$ if for any measurable $\varphi$ with $|{\mathbf{\text{E}}}_{\pi }\left(\varphi \right)|<\infty$,

${\stackrel{^}{E}}_{H}\left(\varphi \right)=\frac{{\sum }_{j=1}^{H}{w}^{\left(j,H\right)}\varphi \left({\vartheta }^{\left(j,H\right)}\right)}{{\sum }_{j=1}^{H}{w}^{\left(j,H\right)}}\to {\mathbf{\text{E}}}_{\pi }\left(\varphi \right)·$

A sequential Monte Carlo algorithm (a particle filter) produces recursively (using mutation-correction-resampling scheme) a sequence of particle systems which target a sequence of distributions ${\pi }_{t}$ on ${{\Theta }}_{t}$. In the Bayes estimation problems ${{\Theta }}_{t}={\Theta }$ is the parameter space and ${\pi }_{t}$ is an a posteriori distribution of the parameter $\vartheta$ given the sample of size $t$. In the state-space filtering or smoothing ${{\Theta }}_{t}$ is the space of states trajectories and ${\pi }_{t}$ is the conditional distribution of the trajectory given the data.

The author obtains conditions for the central limit theorem of the form $\sqrt{H}\left({\stackrel{^}{E}}_{H}\left(\varphi \right)-{\mathbf{\text{E}}}_{\pi }\left(\varphi \right)\right)⇒N\left(0,{V}_{t}\left(\varphi \right)\right)$ where ${V}_{t}\left(\varphi \right)$ is described using recursive formulae. These conditions hold for many of sequential Monte Carlo algorithms including the resample-move algorithm and the residual resampling scheme. Asymptotics of ${V}_{t}\left(\varphi \right)$ as $t\to \infty$ are investigated for Bayesian problems.

##### MSC:
 65C05 Monte Carlo methods 62F15 Bayesian inference 60F05 Central limit and other weak theorems 62L10 Sequential statistical analysis