An Adaptive Metropolis Algorithm

Abstract

A proper choice of a proposal distribution for Markov chain Monte Carlo methods, for example for the Metropolis±Hastings algorithm, is well known to be a crucial factor for the convergence of the algorithm. In this paper we introduce an adaptive Metropolis (AM) algorithm, where the Gaussian proposal distribution is updated along the process using the full information cumulated so far. Due to the adaptive nature of the process, the AM algorithm is non-Markovian, but we establish here that it has the correct ergodic properties. We also include the results of our numerical tests, which indicate that the AM algorithm competes well with traditional Metropolis±Hastings algorithms, and demonstrate that the AM algorithm is easy to use in practical computation.

Markov Chain Monte Carlo Algorithms: Theory and Practice

“…Similar success has been found in other high-dimensional examples (see e.g. [14,26]), and we expect that adaptive MCMC will be used more often in the years ahead.…”

“…A number of recent papers [14,1,3,25,26,34,4,2] have considered the possibility of having the computer modify the Markov chain transitions while the chain runs, in an effort to seek better convergence. This raises a number of theoretical and practical issues, which we now discuss.…”

“…Now, the target covariance Σ is generally unknown, but it can be approximated by Σ n , the empirical covariance of the first n iterations of the Markov chain. This suggests [14,26] an adaptive MCMC algorithm with proposal distribution at the n th iteration given by the mixture distribution…”

Identification of the viscoelastic parameters of a polymer model by the aid of a MCMC method

“…For a more detailed summary of the basic concepts and algorithms of Bayesian parameter estimation and MCMC methods, see (Liu et al, 2013;Solonen and Haario, 2012). In this study, up-to-date adaptive computational schemes are employed in order to make the simulations as effective as possible (Haario et al, 2001). Furthermore, for the Adaptive Metropolis algorithm used here, see the readily available MCMC toolbox for Matlab, which provides the necessary MCMC tools for computational analysis, including several example problems (Laine, 2013).…”

Hydraulic–hydrologic model for water resources management of the Zambezi basin

“…By doing so, it draws from the particular strengths of the best performing algorithms for each given problem, potentially reaching better results more efficiently. Particularly, solutions are adaptively changed based on the shape of the fitness landscape using four optimization methods: (i) non-dominated sorted genetic algorithm-II (Deb et al 2002) optimization (Kennedy & Eberhart 1995), (iii) adaptive Metropolis search (Haario et al 2001) and (iv) differential evolution (Storn & Price 1997). The population of parameter sets evolves based on the results of the previous populations.…”