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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 $(\vartheta^{(j,H)},w^{(j,H)})_{j\le H}$ where $\vartheta^{(j,H)}\in\Theta$ are “particles” and $w^{(j,H)}>0$ are their “weights”. The system targets a distribution $\pi$ on $\Theta$ if for any measurable $\varphi$ with $\vert \text{\bf E}_\pi (\varphi)\vert <\infty$, $$\hat E_H(\varphi)= { \sum_{j=1}^H w^{(j,H)}\varphi(\vartheta^{(j,H)}) \over \sum_{j=1}^H w^{(j,H)} } \to \text{\bf E}_\pi (\varphi).$$ 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 $\sqrt{H}(\hat E_H(\varphi)-\text{\bf E}_\pi (\varphi)) \Rightarrow N(0,V_t(\varphi))$ where $V_t(\varphi)$ 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 $V_t(\varphi)$ 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
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##### References:
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