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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:

60F05 Central limit and other weak theorems
65C40 Numerical analysis or methods applied to Markov chains
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