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An adaptive Metropolis algorithm. (English) Zbl 0989.65004
The choice of the proposal distribution for the random walk Metropolis algorithms is known to be a crucial factor for the convergence of the algorithm. The authors introduce an adaptive variant of the Metropolis algorithm where the proposal distribution is updated using the information about the target distribution obtained so far. This adapted proposal distribution ${q}_{t}\left(·|{X}_{0},\cdots ,{X}_{t-1}\right)$ is a Gaussian distribution with mean at the current point ${X}_{t-1}$ and covariance ${C}_{t}$ being a function of ${X}_{0},\cdots ,{X}_{t-1}$. Although the adaptive algorithm is non-Markovian, it is possible to prove that it has the correct ergodic properties if the target distribution has a bounded support in ${ℝ}^{d}$. The authors report results of numerical tests, which indicate that the adaptive algorithm competes well with the traditional Metropolis-Hastings algorithms.

##### MSC:
 65C40 Computational Markov chains (numerical analysis) 60J22 Computational methods in Markov chains
MCMC