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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 $\mu$, $\pi (dx) =\pi (x)\,d\mu (x)$, where $\pi (x)$ is density. Let $\pi(f) = \int f(x) \pi(dx).$ If we can obtain an i.i.d. sample $x_1, \dots, x_N$ simulated from $\pi$, then $N^{-1} \sum_{i=1}^N f(x_i) = \widehat{\pi}_N (f)$ converges to $\pi (f)$ as $N \to \infty$ with probability one and we can approximate $\pi (f)$ by $\pi_N (f).$ As the normalizing constant of the distribution $\pi$ is unknown, it is not possible to use the estimator $\widehat{\pi}_N (f)$ directly. The authors propose an algorithm for the estimation $\pi (f).$ 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:
60F05Central limit and other weak theorems
65C40Computational Markov chains (numerical analysis)
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