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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 \((\vartheta^{(j,H)},w^{(j,H)})_{j\leq H}\) 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 \(| \mathbf{E}_\pi (\varphi)| <\infty\), \[ \hat E_H(\varphi)= { \sum_{j=1}^H w^{(j,H)}\varphi(\vartheta^{(j,H)}) \over \sum_{j=1}^H w^{(j,H)} } \to \mathbf{E}_\pi (\varphi). \] 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 \(\sqrt{H}(\hat E_H(\varphi)-\mathbf{E}_\pi (\varphi)) \Rightarrow N(0,V_t(\varphi))\) 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.

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