A critical issue for users of Markov Chain Monte Carlo (MCMC) methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest. Research into methods of computing theoretical convergence bounds holds promise for the future but currently has yielded relatively little that is of practical use in applied work. Consequently, most MCMC users address the convergence problem by applying diagnostic tools to the output produced by running their samplers. After giving a brief overview of the area, we provide an expository review of thirteen convergence diagnostics, describing the theoretical basis and practical implementation of each. We then compare their performance in two simple models and conclude that all the methods can fail to detect the sorts of convergence failure they were designed to identify. We thus recommend a combination of strategies aimed at evaluating and accelerating MCMC sampler conver...

Most probabilistic inference algorithms are specified and processed on a propositional level. In the last decade, many proposals for algorithms accepting first-order specifications have been presented, but in the inference stage they still operate on a mostly propositional representation level. [Poole, 2003] presented a method to perform inference directly on the first-order level, but this method is limited to special cases. In this paper we present the first exact inference algorithm that operates directly on a first-order level, and that can be applied to any first-order model (specified in a language that generalizes undirected graphical models). Our experiments show superior performance in comparison with propositional exact inference. 1

This paper is organised as follows. In Section 2, we present an over-simplified version of a convergence diagnostic, and study analytically its performance on certain simple Markov chains. We restrict ourselves primarily to chains which in fact produce i.i.d. samples from

The last ten years have witnessed the development of sampling frameworks that permit the construction of Markov chains which simultaneously traverse both parameter and model space. In this time substantial methodological progress has been made. In this article we present a survey of the current state of the art and evaluate some of the most recent advances in this field. We also discuss future research perspectives in the context of the drive to develop sampling mechanisms with high degrees of both efficiency and automation. 1

We introduce standardised building blocks designed to be used with variational Bayesian learning. The blocks include Gaussian variables, summation, multiplication, nonlinearity, and delay. A large variety of latent variable models can be constructed from these blocks, including variance models and nonlinear modelling, which are lacking from most existing variational systems. The introduced blocks are designed to fit together and to yield e#cient update rules. Practical implementation of various models is easy thanks to an associated software package which derives the learning formulas automatically once a specific model structure has been fixed. Variational Bayesian learning provides a cost function which is used both for updating the variables of the model and for optimising the model structure. All the computations can be carried out locally, resulting in linear computational complexity. We present

this article, we review some important general methods for Markov chain Monte Carlo

Recent advances in statistical software have led to the rapid diffusion of new methods for modeling longitudinal data. Multilevel (also known as hierarchical or random effects) models for binary outcomes have been generally based on a logistic-normal specification, by analogy with earlier work for normally distributed data. The appropriate application and interpretation of these models remains somewhat unclear, especially when compared with the computationally more straightforward marginal modeling (GEE) approaches. In this paper we pose two interrelated questions. First, what limits should be placed on the interpretation of the coefficients and inferences derived from random effect models involving binary outcomes? Second, what are the minimum diagnostic checks that are required to evaluate whether such random effect models provide appropriate fits to the data? We address these questions by means of an extended case study using data on adolescent smoking from a large cohort study. Bay...

JAGS is a program for Bayesian Graphical modelling which aims for compatibility with Classic BUGS. The program could eventually be developed as an R package. This article explains the motivations for this program, briefly describes the architecture and then discusses some ideas for a vectorized form of the BUGS language.

In the models discussed in this book so far we have assumed the populations from which data has been drawn are hierarchical. This assumption is not always justified. Two main types of nonhierarchical model are considered in this chapter. Cross classified models and multiple membership models. This chapter draws on the work of Rasbash and Goldstein(1996) and Hill and Goldstein(1998) 1 Cross-classified models This section is divided into four parts. In this first part we look at situations in health research that can give rise to a two way cross-classification and suggest some notation to describe this model. In the second part we look at more complicated cross-classified structures and extend the notation. In the third part we describe general rules for the notation construction. In the final part we describe the analysis of an example data set. 1.1 Two way cross-classifications – a basic model. Suppose, we have data on a large number of patients, attending many hospitals and we also know the neighbourhood in which the patient lives and that we regard patient, neighbourhood and hospital all as important sources of variation for the patient level outcome measure we wish to study. Now, typically hospitals will draw patients from many different neighbourhoods and the inhabitants of a neighbourhood will go to many hospitals. No pure hierarchy can be found and patients are said to be contained within a cross-classification of hospitals by neighbourhoods. This can be represented diagrammatically, for the case of twenty patients contained within a cross-classification of three neighbourhoods by five hospitals: Table 1: patients cross classified by hospital and neighbourhood neighbourhood 1 Neighbourhood 2 neighbourhood 3

The accuracy of the Gibbs sampling Markov chain monte carlo procedure was examined for estimating item and person (θ) parameters in the one-parameter logistic model. Four datasets were analyzed using the Gibbs sampling method, conditional maximum likelihood, marginal maximum likelihood, and joint maximum likelihood. Maximum likelihood and expected a posteriori θ estimation methods were used with marginal maximum likelihood estimation of item parameters. Item parameter estimates from the four methods were almost identical; θ estimates from Gibbs sampling were similar to those obtained from the expected a posteriori method. Index terms: Bayesian inference, conditional maximum likelihood, Gibbs sampling, item response theory, joint maximum