In this article, we present a behind-the-scenes look at a Bayesian hierarchical analysis of pathways of exposure to arsenic (a toxic heavy metal) using the Phase I National Human Exposure Assessment Survey carried out in Arizona. Our analysis combines individual-level personal exposure measurements (biomarker and environmental media) with water, soil, and air observations from the ambient environment. We include details of our model-building exercise that involved a combination of exploratory data analysis and substantive knowledge in exposure science. Then we present our strategies for model fitting, which involved piecing together components of the hierarchical model in a systematic fashion to assess issues including parameter identifiability, Bayesian learning, model fit, and convergence diagnostics. We also discuss practical issues of data management and algorithm debugging, especially in the light of missing and censored data. We believe that our presentation of these behind-the-scenes details will be of use to other researchers who build complex Bayesian hierarchical models.

We congratulate the authors on their development of Bayesian hierarchical models that simultaneously overcome many of the challenges inherent to analyzing exposure pathways to Arsenic. Drs. Craigmile, Calder, Li, Paul and Cressie (herein referenced by CCLPC) address a range of obstacles with a sound mix of sophisticated models, useful plots and common sense. Their paper is well written throughout, making important illustrative, methodological and applied contributions. We thank the editor for inviting us to discuss this stimulating paper, and will begin by calling special attention to some particular highlights of the analysis by CCLPC. After which, we will mention a few aspects we believe can be further improved.

We want to estimate the parameters governing a continuous time Markov switching model given observations at discrete times only. For parameter estimation in a setting with continuous time and a latent state process, using MCMC methods two approaches are common: Using time-discretization and augmenting the unknowns with the (then discrete) state process, or working in continuous time and augmenting with the full state process. In this paper, we combine useful aspects of both approaches. On the one hand, we are inspired by the discretization, where filtering for the state process is possible, on the other hand, we catch attractive features of the continuous time method, like exact estimation (i.e. no discretization error) and direct estimation of the generator matrix rather than the transition matrix. This is achieved by taking not the whole state process for data augmentation but only the states at observation times. Using results on the distribution of occupation times in Markov processes, it is possible to compute the complete data likelihood exactly. We obtain a sampler that works more robustly and more accurately especially for fast switching in the state process.

The selection of variables in regression problems has occupied the minds of many statisticians. Several Bayesian variable selection methods have been developed, and we concentrate on the following methods: Kuo & Mallick, Gibbs Variable Selection (GVS), Stochastic Search Variable Selection (SSVS), adaptive shrinkage with Jeffreys' prior or a Laplacian prior, and reversible jump MCMC. We review these methods, in the context of their different properties. We then implement the methods in BUGS, using both real and simulated data as examples, and investigate how the different methods perform in practice. Our results suggest that SSVS, reversible jump MCMC and adaptive shrinkage methods can all work well, but the choice of which method is better will depend on the priors that are used, and also on how they are implemented.

Bayesian analysis incorporates different sources of information into a single analysis through Bayes theorem. When one or more of the sources of information are suspect (e.g., if the model assumed for the information is viewed as quite possibly being significantly flawed), there can be a concern that Bayes theorem allows this suspect information to overly influence the other sources of information. We consider a variety of situations in which this arises, and give methodological suggestions for dealing with the problem.

After consideration of some pedagogical examples of the phenomenon, we focus on the interface of statistics and the development of complex computer models of processes. Three testbed computer models are considered, in which this type of issue arises.

It is argued that the posterior predictive distribution for the binomial and multinomial distributions, when viewed via a hypergeometric-like representation, suggests the uniform prior on the parameters for these models. The argument is supported by studying variations on an example by Fisher, and complements Bayes' original argument for a uniform prior predictive distribution for the binomial. The fact that both arguments lead to invariance under transformation is also discussed.

We develop a Bayesian procedure for analyzing stationary long-range dependent processes. Specifically, we consider the fractional exponential model (FEXP) to estimate the memory parameter of a stationary long-memory Gaussian time series. In particular, we propose a hierarchical Bayesian model and make it fully adaptive by imposing a prior distribution on the model order. Further, we describe a reversible jump Markov chain Monte Carlo algorithm for variable dimension estimation and show that, in our context, the algorithm provides a reasonable method of model selection (within each repetition of the chain). Therefore, through an application of Bayesian model averaging, we incorporate all possible models from the FEXP class (up to a given finite order). As a result we reduce the underestimation of uncertainty at the model-selection stage as well as achieve better estimates of the long memory parameter. Additionally, we establish Bayesian consistency of the memory parameter under mild conditions on the data process. Finally, through simulation and the analysis of two data sets, we demonstrate the effectiveness of our approach.