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...

Abstract: Let pi(w),i =1, 2, be two densities with common support where each density is known up to a normalizing constant: pi(w) =qi(w)/ci. We have draws from each density (e.g., via Markov chain Monte Carlo), and we want to use these draws to simulate the ratio of the normalizing constants, c1/c2. Such a computational problem is often encountered in likelihood and Bayesian inference, and arises in fields such as physics and genetics. Many methods proposed in statistical and other literature (e.g., computational physics) for dealing with this problem are based on various special cases of the following simple identity: c1 c2 = E2[q1(w)α(w)] E1[q2(w)α(w)]. Here Ei denotes the expectation with respect to pi (i =1, 2), and α is an arbitrary function such that the denominator is non-zero. A main purpose of this paper is to provide a theoretical study of the usefulness of this identity, with focus on (asymptotically) optimal and practical choices of α. Using a simple but informative example, we demonstrate that with sensible (not necessarily optimal) choices of α, we can reduce the simulation error by orders of magnitude when compared to the conventional importance sampling method, which corresponds to α =1/q2. We also introduce several generalizations of this identity for handling more complicated settings (e.g., estimating several ratios simultaneously) and pose several open problems that appear to have practical as well as theoretical value. Furthermore, we discuss related theoretical and empirical work.

We will apply the method of blocking Gibbs sampling to a problem of great importance and complexity -- linkage analysis. Blocking Gibbs combines exact local computations with Gibbs sampling in a way that complements the strengths of both. The method is able to handle problems with very high complexity such as linkage analysis in large pedigrees with many loops; a task that no other known method is able to handle. New developments of the method are outlined, and it is applied to a highly complex linkage problem. Keywords: Bayesian network, junction tree, pedigree analysis, Markov chain Monte Carlo, Gibbs sampling, loops, inbreeding 1 Introduction For linkage analysis - the problem of estimating the relative positions of the genes on the chromosomes - many methods have been developed over recent years. Fast and exact methods for computation in Bayesian networks (e.g., pedigrees) (Cannings, Thompson & Skolnick 1976; Pearl 1986; Lauritzen & Spiegelhalter 1988; Shenoy & Shafer 1990; Lauri...

In recent years, many investigators have proposed Gibbs prior models to regularize images reconstructed from emission computed tomography data. Unfortunately, hyperparameters used to specify Gibbs priors can greatly influence the degree of regularity imposed by such priors, and as a result, numerous procedures have been proposed to estimate hyperparameter values from observed image data. Many of these procedures attempt to maximize the joint posterior distribution on the image scene. To implement these methods, approximations to the joint posterior densities are required, because the dependence of the Gibbs partition function on the hyperparameter values is unknown. In this paper, we use recent results in Markov Chain Monte Carlo sampling to estimate the relative values of Gibbs partition functions, and using these values, sample from joint posterior distributions on image scenes. This allows for a fully Bayesian procedure which does not fix the hyperparameters at some estimated or spe...

Abstract. Ratios of normalizing constants for two distributions are needed in both Bayesian statistics, where they are used to compare models, and in statistical physics, where they correspond to differences in free energy. Two approaches have long been used to estimate ratios of normalizing constants. The ‘simple importance sampling ’ (SIS) or ‘free energy perturbation ’ method uses a sample drawn from just one of the two distributions. The ‘bridge sampling ’ or ‘acceptance ratio ’ estimate can be viewed as the ratio of two SIS estimates involving a bridge distribution. For both methods, difficult problems must be handled by introducing a sequence of intermediate distributions linking the two distributions of interest, with the final ratio of normalizing constants being estimated by the product of estimates of ratios for adjacent distributions in this sequence. Recently, work by Jarzynski, and independently by Neal, has shown how one can view such a product of estimates, each based on simple importance sampling using a single point, as an SIS estimate on an extended state space. This ‘Annealed Importance Sampling ’ (AIS) method produces an exactly unbiased estimate for the ratio of normalizing constants even when the Markov transitions used do not reach equilibrium. In this paper, I show how a corresponding ‘Linked Importance Sampling ’ (LIS) method can be constructed in which the estimates for individual ratios are similar to bridge sampling estimates. As a further

This article describes a procedure for defining a posterior distribution on the value of a normalizing constant or ratio of normalizing constants using output from Monte Carlo simulation experiments. The resulting posterior distribution provides a simple diagnostic for assessing the adequacy of a simulation experiment for estimating these quantities, and is particularly useful in cases for which standard estimators perform poorly, since in such situations asymptotic properties of standard diagnostics are unlikely to hold. Keywords: Marginal likelihood, partition function, Markov chain Monte Carlo, Ising model, fl coupling. 1 Introduction This article describes a simulation-based method for computing a posterior distribution on either a single normalizing constant or a ratio of normalizing constants. The method relies on a coupling argument to define two sequences of Bernoulli random variables whose success probabilities, given the true values of the normalizing constants, are...

In this paper I am presenting an overview on several topics related to nonequilibrium fluctuations in small systems. I start with a general discussion about fluctuation theorems and applications to physical examples extracted from physics and biology: a bead in an optical trap and single molecule force experiments. Next I present a general discussion on path thermodynamics and consider distributions of work/heat fluctuations as large deviation functions. Then I address the topic of glassy dynamics from the perspective of nonequilibrium fluctuations due to small cooperatively rearranging regions. Finally, I

Nested sampling is a new Monte Carlo method by Skilling [1] intended for general Bayesian computation. Nested sampling provides a robust alternative to annealing-based methods for computing normalizing constants. It can also generate estimates of other quantities such as posterior expectations. The key technical requirement is an ability to draw samples uniformly from the prior subject to a constraint on the likelihood. We provide a demonstration with the Potts model, an undirected graphical model.

Abstract not found

ABSTRACT A three-dimensional model structure of a complex formed by a G-protein-coupled receptor (GPCR) and an agonist ligand is probed and refined using molecular-dynamics simulations and free energy calculations in a realistic environment. The model of the human receptor of cholecystokinin associated to agonist ligand CCK9 was obtained from a synergistic procedure combining site-directed mutagenesis experiments and in silico modeling. The 31-ns molecular-dynamics simulation in an explicit membrane environment indicates that both the structure of the receptor and its interactions with the ligand are robust. Whereas the secondary structure of the a-helix bundle is well preserved, the region of the intracellular loops exhibits a significant flexibility likely to be ascribed to the absence of G-protein subunits in the model. New insight into the structural features of the binding pocket is gained, in particular, the interplay of the ligand with both the receptor and internal water molecules. Water-mediated interactions are shown to participate in the binding, hence, suggesting additional site-directed mutagenesis experiments. Accurate free energy calculations on mutated ligands provide differences in the receptor-ligand binding affinity, thus offering a direct, quantitative comparison to experiment. We propose that this detailed consistencychecking procedure be used as a routine refinement step of in vacuo GPCR models, before further investigation and application to structure-based drug design.