Adaptive Metropolis–Hastings sampling using reversible dependent mixture proposals
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This article develops a general-purpose adaptive sampler for sampling from a high-dimensional and/or multimodal target. The adaptive sampler is based on reversible proposal densities each of which has a mixture of multivariate \(t\) densities as its invariant density. The reversible proposals are a combination of independent and correlated components that allow the sampler to traverse the sample space efficiently as well as allowing the sampler to keep moving and exploring the sample space locally. We employ a two-chain approach, using a trial chain to adapt the proposal in the main chain. Convergence of the main chain and a strong law of large numbers are obtained under checkable conditions, and without imposing a diminishing adaptation condition. The mixtures of multivariate \(t\) densities are fitted by an efficient variational approximation algorithm in which the number of components is determined automatically. The performance of the sampler is evaluated using simulated and real examples. Our framework is quite general and can handle reversible proposal densities whose invariant densities are mixtures other than \(t\) mixtures.
KeywordsErgodic convergence Markov chain Monte Carlo Metropolis-within Gibbs sampling Multivariate \(t\) mixtures Simulated annealing Variational approximation
The research of Minh-Ngoc Tran and Robert Kohn was partially supported by Australian Research Council grant DP0667069. The authors thank the referee and the Associate Editor for their comments which improved the presentation and technical content of the paper.
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