While Bayes' theorem has a 250-year history, and the method of inverse probability that flowed from it dominated statistical thinking into the twentieth century, the adjective "Bayesian" was not part of the statistical lexicon until relatively recently. This paper provides an overview of key Bayesian developments, beginning with Bayes' posthumously published 1763 paper and continuing up through approximately 1970, including the period of time when "Bayesian" emerged as the label of choice for those who advocated Bayesian methods.

Understanding spatial patterns of species diversity and the distributions of individual species is a consuming problem in biogeography and conservation. The Cape Floristic Region (CFR) of South Africa is a global hotspot of diversity and endemism, and the Protea Atlas Project, with some 60,000 site records across the region, provides an extraordinarily rich data set to analyze biodiversity patterns. Analysis for the region is developed at the spatial scale of one minute grid-cells (~37,000$ cells total for the region). We report on results for 40 species of a flowering plant family Proteaceae (of about 330 in the CFR) for a defined subregion.

Using a Bayesian framework, we develop a two stage, spatially explicit, hierarchical logistic regression. Stage one models the suitability or potential presence for each species at each cell, given species attributes along with grid cell (site-level) climate, precipitation, topography and geology data using species-level coefficients, and a spatial random effect. The second level of the hierarchy models, for each species, observed presence$/$absence at a sampling site through a conditional specification of the probability of presence at an arbitrary location in the grid cell given that the location is suitable. Because the atlas data are not evenly distributed across the landscape, grid cells contain variable numbers of sampling localities. Indeed, some grid cells are entirely unsampled; others have been transformed by human intervention (agriculture, urbanization) such that none of the species are there though some may have the potential to be present in the absence of disturbance. Thus the modeling takes the sampling intensity at each site into account by assuming that the total number of times that a particular species was observed within a site follows a binomial distribution.

In fact, a range of models can be examined incorporating different first and second stage specifications. This necessitates model comparison in a misaligned multilevel setting. All models are fitted using MCMC methods. A "best" model is selected. Parameter summaries offer considerable insight. In addition, results are mapped as the model-estimated potential presence for each species across the domain. This probability surface provides an alternative to customary empirical "range of occupancy" displays. Summing yields the predicted species richness over the region. Summaries of the posterior for each environmental coefficient show which variables are most important in explaining species presence. Other biodiversity measures emerge as model unknowns. A considerable range of inference is available. We illustrate with only a portion of the analyses we have conducted, noting that these initial results describe biogeographical patterns over the modeled region remarkably well.

The study of genetics continues to advance dramatically with the development of microarray technology. In light of the advancements, interesting statistical challenges have arisen. Given that only one observation can be made from each gene on a single array, statisticians are faced with three issues: analysis with more genes than arrays, separating true differential expression from noise, and multiple hypothesis testing for regulation. Within this study, we model the expression of 1185 genes simultaneously in response to five chemical constituents of particulate matter; arsenic, iron, nickel, vanadium, and zinc. Taking advantage of a hierarchical Bayesian mixture model with latent variables, we compare multiple treatments to a control and estimate noise across arrays without assuming equal treatment means for housekeeping genes. To account for model uncertainty and hyperparameter specification, model averaging, MCMC, and Rao-Blackwell estimation are utilized.

Dirichlet process (DP) mixture models are the cornerstone of nonparametric Bayesian statistics, and the development of Monte-Carlo Markov chain (MCMC) sampling methods for DP mixtures has enabled the application of nonparametric Bayesian methods to a variety of practical data analysis problems. However, MCMC sampling can be prohibitively slow, and it is important to explore alternatives. One class of alternatives is provided by variational methods, a class of deterministic algorithms that convert inference problems into optimization problems (Opper and Saad 2001; Wainwright and Jordan 2003). Thus far, variational methods have mainly been explored in the parametric setting, in particular within the formalism of the exponential family (Attias 2000; Ghahramani and Beal 2001; Blei et al. 2003). In this paper, we present a variational inference algorithm for DP mixtures. We present experiments that compare the algorithm to Gibbs sampling algorithms for DP mixtures of Gaussians and present an application to a large-scale image analysis problem.

In this paper we discuss auxiliary variable approaches to Bayesian binary and multinomial regression. These approaches are ideally suited to automated Markov chain Monte Carlo simulation. In the first part we describe a simple technique using joint updating that improves the performance of the conventional probit regression algorithm. In the second part we discuss auxiliary variable methods for inference in Bayesian logistic regression, including covariate set uncertainty. Finally, we show how the logistic method is easily extended to multinomial regression models. All of the algorithms are fully automatic with no user set parameters and no necessary Metropolis-Hastings accept/reject steps.

Bayesian robustness modelling using heavy-tailed distributions provides a flexible approach to resolving problems of conflicts between the data and prior distributions. See Dawid (1973) and O'Hagan (1979, 1988, 1990), who provided sufficient conditions on the distributions in the model in order to reject the conflicting data or the prior distribution in favour of the other source of information. However, the literature has almost concentrated exclusively on robustness of the posterior distribution of location parameters; little attention has been given to scale parameters. In this paper we propose a new approach for Bayesian robustness modelling, in which we use the class of regularly varying distributions. Regular variation provides a very natural description of tail thickness in heavy-tailed distributions. Using regular variation theory, we establish sufficient conditions in the pure scale parameter structure under which is possible to resolve conflicts amongst the sources of information. We also note some important differences between the scale and the location parameters cases. Finally, we obtain new conditions in the pure location parameter structure which may be easier to verify than those proposed by Dawid and O'Hagan.