A. F. M. Smith

Sampling-Based Approaches to Calculating Marginal Densities

Abstract Stochastic substitution, the Gibbs sampler, and the sampling-importance-resampling algorithm can be viewed as three alternative sampling- (or Monte Carlo-) based approaches to the calculation of numerical estimates of marginal probability distributions. The three approaches will be reviewed, compared, and contrasted in relation to various joint probability structures frequently encountered in applications. In particular, the relevance of the approaches to calculating Bayesian posterior densities for a variety of structured models will be discussed and illustrated.

Statistical analysis of finite mixture distributions

Statistical Problems. Applications of Finite Mixture Models. Mathematical Aspects of Mixtures. Learning About the Parameters of a Mixture. Learning About the Components of a Mixture. Sequential Problems and Procedures.

Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling

Abstract The use of the Gibbs sampler as a method for calculating Bayesian marginal posterior and predictive densities is reviewed and illustrated with a range of normal data models, including variance components, unordered and ordered means, hierarchical growth curves, and missing data in a crossover trial. In all cases the approach is straightforward to specify distributionally and to implement computationally, with output readily adapted for required inference summaries.

Bayesian Statistics Without Tears: A Sampling-Resampling Perspective

Abstract Even to the initiated, statistical calculations based on Bayess Theorem can be daunting because of the numerical integrations required in all but the simplest applications. Moreover, from a teaching perspective, introductions to Bayesian statistics—if they are given at all—are circumscribed by these apparent calculational difficulties. Here we offer a straightforward sampling-resampling perspective on Bayesian inference, which has both pedagogic appeal and suggests easily implemented calculation strategies.

Hierarchical Bayesian Analysis of Changepoint Problems

SUMMARY A general approach to hierarchical Bayes changepoint models is presented. In particular, desired marginal posterior densities are obtained utilizing the Gibbs sampler, an iterative Monte Carlo method. This approach avoids sophisticated analytic and numerical high dimensional integration procedures. We include an application to changing regressions, changing Poisson processes and changing Markov chains. Within these contexts we handle several previously inaccessible problems.

Bayesian Analysis of Constrained Parameter and Truncated Data Problems Using Gibbs Sampling

Abstract : Bayesian analysis of constrained parameter and truncated data problems is complicated by the seeming need for, typically multidimensional, numerical integrations over awkwardly defined regions. This paper illustrates how the Gibbs sampler approach to Bayesian calculation (Gelfand and Smith, 1990) avoids these difficulties and leads to straightforwardly implemented procedures, even for apparently very complicated model forms.

Automatic Bayesian curve fitting

A method of estimating a variety of curves by a sequence of piecewise polynomials is proposed, motivated by a Bayesian model and an appropriate summary of the resulting posterior distribution. A joint distribution is set up over both the number and the position of the knots defining the piecewise polynomials. Throughout we use reversible jump Markov chain Monte Carlo methods to compute the posteriors. The methodology has been successful in giving good estimates for ‘smooth’ functions (i.e. continuous and differentiable) as well as functions which are not differentiable, and perhaps not even continuous, at a finite number of points. The methodology is extended to deal with generalized additive models.

Bayesian analysis of linear and non-linear population models by using the Gibbs sampler

Abstract : A fully Bayesian analysis of linear and nonlinear population models has previously been unavailable, as a consequence of the seeming impossibility of performing the necessary numerical Integrations in the complex multi- parameter structures typically arising in such models. It is demonstrated that, for a variety of linear and nonlinear population models, a fully Bayesian analysis can be implemented in a straightforward manner using the Gibbs sampler. The approach is illustrated with examples involving challenging problems of outliers and mean-variance relationships in population modelling.

Efficient generation of random variates via the ratio-of-uniforms method

Improvements to the conventional ratio-of-uniforms method for random variate generation are proposed. A generalized radio-of-uniforms method is introduced, and it is demonstrated that relocation of the required density via the mode can greatly improve the computational efficiency of the method. We describe a multivariate version of the basic method and summarize a general strategy for efficient ratio-of-uniforms generation. Illustrative examples are given.

Modeling and Monitoring Biomedical Time Series

Abstract A framework for modeling and monitoring medical time series is developed, based on extensions to the linear dynamic model. The approach is designed to provide for on-line, recursive updating and inference for time series that are subject to several forms of potential discontinuous change, and that may have missing values. The models and methods introduced are illustrated with three biomedical series.