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Convergence of adaptive mixtures of importance sampling schemes. (English) Zbl 1132.60022
Let $\pi$ be a probability distribution, $\pi$ is dominated by a reference measure $\mu$, $\pi \left(dx\right)=\pi \left(x\right)\phantom{\rule{0.166667em}{0ex}}d\mu \left(x\right)$, where $\pi \left(x\right)$ is density. Let $\pi \left(f\right)=\int f\left(x\right)\pi \left(dx\right)·$ If we can obtain an i.i.d. sample ${x}_{1},\cdots ,{x}_{N}$ simulated from $\pi$, then ${N}^{-1}{\sum }_{i=1}^{N}f\left({x}_{i}\right)={\stackrel{^}{\pi }}_{N}\left(f\right)$ converges to $\pi \left(f\right)$ as $N\to \infty$ with probability one and we can approximate $\pi \left(f\right)$ by ${\pi }_{N}\left(f\right)·$ As the normalizing constant of the distribution $\pi$ is unknown, it is not possible to use the estimator ${\stackrel{^}{\pi }}_{N}\left(f\right)$ directly. The authors propose an algorithm for the estimation $\pi \left(f\right)·$ The authors derive sufficient convergence conditions for adaptive mixtures of population Monte Carlo algorithms and show that Rao-Blackwellized asymptotically achieve an optimum in terms of a Kullback divergence criterion.
MSC:
 60F05 Central limit and other weak theorems 65C40 Computational Markov chains (numerical analysis)