Simulated annealing --- moving from a tractable distribution to a distribution of interest via a sequence of intermediate distributions --- has traditionally been used as an inexact method of handling isolated modes in Markov chain samplers. Here, it is shown how one can use the Markov chain transitions for such an annealing sequence to define an importance sampler. The Markov chain aspect allows this method to perform acceptably even for high-dimensional problems, where finding good importance sampling distributions would otherwise be very difficult, while the use of importance weights ensures that the estimates found converge to the correct values as the number of annealing runs increases. This annealed importance sampling procedure resembles the second half of the previously-studied tempered transitions, and can be seen as a generalization of a recently-proposed variant of sequential importance sampling. It is also related to thermodynamic integration methods for estimating ratios...

Computing (ratios of) normalizing constants of probability models is a fundamental computational problem for many statistical and scientific studies. Monte Carlo simulation is an effective technique, especially with complex and high-dimensional models. This paper aims to bring to the attention of general statistical audiences of some effective methods originating from theoretical physics and at the same time to explore these methods from a more statistical perspective, through establishing theoretical connections and illustrating their uses with statistical problems. We show that the acceptance ratio method and thermodynamic integration are natural generalizations of importance sampling, which is most familiar to statistical audiences. The former generalizes importance sampling through the use of a single "bridge" density and is thus a case of bridge sampling in the sense of Meng and Wong (1996). Thermodynamic integration, which is also known in the numerical analysis literature as Oga...

. I present a new Markov chain sampling method appropriate for distributions with isolated modes. Like the recently-developed method of "simulated tempering", the "tempered transition" method uses a series of distributions that interpolate between the distribution of interest and a distribution for which sampling is easier. The new method has the advantage that it does not require approximate values for the normalizing constants of these distributions, which are needed for simulated tempering, and can be tedious to estimate. Simulated tempering performs a random walk along the series of distributions used. In contrast, the tempered transitions of the new method move systematically from the desired distribution, to the easily-sampled distribution, and back to the desired distribution. This systematic movement avoids the inefficiency of a random walk, an advantage that unfortunately is cancelled by an increase in the number of interpolating distributions required. Because of this, the sa...

In this paper we develop tools for analyzing the rate at which a reversible Markov chain converges to stationarity. Our techniques are useful when the Markov chain can be decomposed into pieces which are themselves easier to analyze. The main theorems relate the spectral gap of the original Markov chains to the spectral gap of the pieces. In the first case the pieces are restrictions of the Markov chain to subsets of the state space; the second case treats a Metropolis-Hastings chain whose equilibrium distribution is a weighted average of equilibrium distributions of other Metropolis-Hastings chains on the same state space.

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Recent developments in statistics now allow maximum likelihood estimators for the parameters of Markov Random Fields to be constructed. We detail the theory required, and present an algorithm which is easily implemented and practical in terms of computation time. We demonstrate this algorithm on three MRF models the standard Potts model, an inhomogeneous variation of the Potts model, and a long-range interaction model, better adapted to modeling real-world images. We estimate the parameters from a synthetic and a real image, and then resynthesise the models to demonstrate which features of the image have been captured by the model. Segmentations are computed based on the estimated parameters and conclusions drawn.

We develop a framework for learning generic, expressive image priors that capture the statistics of natural scenes and can be used for a variety of machine vision tasks. The approach provides a practical method for learning high-order Markov random field (MRF) models with potential functions that extend over large pixel neighborhoods. These clique potentials are modeled using the Product-of-Experts framework that uses nonlinear functions of many linear filter responses. In contrast to previous MRF approaches all parameters, including the linear filters themselves, are learned from training data. We demonstrate the capabilities of this Field-of-Experts model with two example applications, image denoising and image inpainting, which are implemented using a simple, approximate inference scheme. While the model is trained on a generic image database and is not tuned toward a specific application, we obtain results that compete with specialized techniques.

Markov random fields are used extensively in modelbased approaches to image segmentation and, under the Bayesian paradigm, are implemented through Markov chain Monte Carlo (MCMC) methods. In this paper, we describe a class of such models (the double Markov random field) for images composed of several textures, which we consider to be the natural hierarchical model for such a task. We show how several of the Bayesian approaches in the literature can be viewed as modifications of this model, made in order to make MCMC implementation possible. From a simulation study, conclusions are made concerning the performance of these modified models.

“Extended Ensemble Monte Carlo ” is a generic term that indicates a set of algorithms which are now popular in a variety of fields in physics and statistical information processing. Exchange Monte Carlo (Metropolis-Coupled Chain, Parallel Tempering), Simulated Tempering (Expanded Ensemble Monte Carlo), and Multicanonical Monte Carlo (Adaptive Umbrella Sampling) are typical members of this family. Here we give a cross-disciplinary survey of these algorithms with special emphasis on the great flexibility of the underlying idea. In Sec. 2, we discuss the background of Extended Ensemble Monte Carlo. In Sec. 3, 4 and 5, three types of the algorithms, i.e., Exchange Monte Carlo, Simulated Tempering, Multicanonical Monte Carlo, are introduced. In Sec. 6, we give an introduction to Replica Monte Carlo algorithm by Swendsen and Wang. Strategies for the construction of special-purpose extended ensembles are discussed in Sec. 7. We stress

We introduce a new sampling algorithm, the equi-energy sampler, for efficient statistical sampling and estimation. Complementary to the widely used temperature-domain methods, the equi-energy sampler, utilizing the temperature–energy duality, targets the energy directly. The focus on the energy function not only facilitates efficient sampling, but also provides a powerful means for statistical estimation, for example, the calculation of the density of states and microcanonical averages in statistical mechanics. The equi-energy sampler is applied to a variety of problems, including exponential regression in statistics, motif sampling in computational biology and protein folding in biophysics.