Maximum Likelihood Estimation for Spatial Models by Markov Chain Monte Carlo Stochastic Approximation

Abstract

We propose a two-stage algorithm for computing maximum likelihood estimates for a class of spatial models. The algorithm combines Markov chain Monte Carlo methods such as the Metropolis±Hastings±Green algorithm and the Gibbs sampler, and stochastic approximation methods such as the off-line average and adaptive search direction. A new criterion is built into the algorithm so stopping is automatic once the desired precision has been set. Simulation studies and applications to some real data sets have been conducted with three spatial models. We compared the algorithm proposed with a direct application of the classical Robbins±Monro algorithm using Wiebe's wheat data and found that our procedure is at least 15 times faster.

Stochastic Approximation in Monte Carlo Computation

“…Refer to Lai (2003) for an overview on the subject. Recently, it has been used with Markov chain Monte Carlo for solving maximum likelihood estimation problems (Younes, 1988(Younes, , 1999Moyeed and Baddeley, 1991;Gu and Kong, 1998;Gelfand and Banerjee, 1998;Delyon, Lavielle and Moulines, 1999;Gu and Zhu, 2001). The critical difference between SAMC and other stochastic approximation MCMC algorithms is at sample space partitioning.…”

Use of SAMC for Bayesian analysis of statistical models with intractable normalizing constants

“…Andrews and Herzberg, 1985). This data has been analyzed by a number of authors, e.g., Besag (1974), Huang and Ogata (1999), Gu and Zhu (2001) and Liang (2010). Following the previous authors, we subtracted the mean from the data and then fitted the data by the autonormal model.…”

Convergence and convergence rate of stochastic gradient search in the case of multiple and non-isolated extrema

“…The minimization of the negative log-likelihood using stochastic gradient search has a long tradition in statistical inference, system identification and signal and image processing, while the asymptotic properties of the corresponding algorithms have studied in a number of papers (see e.g., [3], [16], [26], [32], [48] and references cited therein). Although the available literature provides a good insight into the asymptotic behavior of the recursive maximum likelihood method, the existing results on the convergence and convergence rate (of algorithm (15)) rely on very restrictive conditions: These results require the negative log-likelihood f (·) to have an isolated minimum θ * and its gradient ∇f (·) to admit representation (5).…”