The deviance information criterion (DIC) introduced by Spiegelhalter et al.(2002) for model assessment and model comparison is directly inspired by linear and generalised linear models, but it is open to different possible variations in the setting of missing data models, depending in particular on whether or not the missing variables are treated as parameters. In this paper, we reassess the criterion for such models and compare different DIC constructions, testing the behaviour of these various extensions in the cases of mixtures of distributions and random effect models.

The Deviance Information Criterion (DIC) for model choice was introduced by Spiegelhalter et al. (2002) as a Bayesian analogue of the Akaike Information Criterion. The aim of DIC is not to identify the "true" probability model, but to find a parsimonious description of the data $Y$ in terms of parameters $\theta$. The parameters are of lower dimension than the data, either because $\theta$ is restricted to a low-dimensional subspace, or because it has a highly structured prior. A penalty function $p_D$ measures the "effective number of parameters" of the model, and this is added to a measure of fit -- the expected deviance -- to give the DIC. Given a set of models, the one with the smallest DIC has the best balance between goodness of fit and model complexity.

DIC has received a mixed reception, as shown by the discussion of Spiegelhalter et al. (2002). On the one hand, it gives a pragmatic solution to the problem of model choice, and is now routinely available in the software WinBUGS (Spiegelhalter et al. 2004). On the other hand, a number of technical and conceptual difficulties with the criterion remain. Celeux et al. (2006) investigate these difficulties in the context of missing data models, and in particular mixture models. They have produced 8 variations on the theme of DIC. Some of these variations address the problem of finding a good "plug-in" estimate of $\theta$, which is necessary for the calculation of the penalty $p_D$. Others are innovations that provide a way of calculating DIC in missing data models, which might otherwise be intractable. I have attempted to classify these criteria according to their level of "focus".

This discussion argues that any difficulty with DIC for missing data is due to DIC being intrinsically a large-sample measure and relying on point estimates. What is missing is not "missing data", but rather a set of coherent principles for DIC itself when the amount of data is not adequate to invoke quadratic approximation for a complex model. The non-uniqueness of data augmentation schemes for any observed-data model also argues for the importance of emphasizing inference "focus" in applying model complexity measures such as DIC. An attempt to bring in more Bayesian "flavor" into DIC also reveals that an insightful explanation is missing: neither pure Bayesian measure nor pure likelihood/sampling measure yield sensible results, but some hybrid ones do.

A major challenge to the statistical analysis of microarray data is the small number of samples — limited by both cost and sample availability — compared to the large number of genes, now soaring into the tens of thousands per experiment. This situation is made even more difficult by the complex nature of the empirical distributions of gene expression measurements and the necessity to limit the number of false detections due to multiple comparisons. This paper introduces a novel Bayesian method for the analysis of comparative experiments performed with oligonucleotide microarrays. Our method models gene expression data by log-normal and gamma distributions with hierarchical prior distributions on the parameters of interest, and uses model averaging to compute the posterior probability of differential expression. An initial approximate Bayesian analysis is used to identify genes that have a large probability of differential expression, and this list of candidate genes is further refined by using stochastic computations. We assess the performance of this method using real data sets and show that it has an almost negligible false positive rate in small sample experiments that leads to a better detection performance.

This paper illustrates a novel hierarchical dynamic Bayesian network modelling the spiking patterns of neuronal ensembles over time. We introduce, at separate model stages, the parameters characterizing the discrete-time spiking process, the unknown structure of the functional connections among the analysed neurons and its dependence on their spatial arrangement. Estimates for all model parameters and predictions for future spiking states are computed under the Bayesian paradigm via the standard Gibbs sampler using a shrinkage prior. The adequacy of the model is investigated by plotting the residuals and by applying the time-rescaling theorem. We analyse a simulated dataset and a set of experimental multiple spike trains obtained from a culture of neurons in vitro. For the latter data, we find that one neuron plays a pivotal role for the initiation of each cycle of network activity and that the estimated network structure significantly depends on the spatial arrangement of the neurons.

A flyer plate experiment involves forcing a plane shock wave through stationary test samples of material and measuring the free surface velocity of the target as a function of time. These experiments are conducted to learn about the behavior of materials subjected to high strain rate environments. Computer simulations of flyer plate experiments are conducted with a two-dimensional hydrodynamic code developed under the Advanced Strategic Computing (ASC) program at Los Alamos National Laboratory. This code incorporates physical models that contain parameters having uncertain values. The objectives of the analyses presented in this paper are to assess the sensitivity of free surface velocity to variations in the uncertain inputs, to constrain the values of these inputs to be consistent with experiment, and to predict free surface velocity based on the constrained inputs. We implement a Bayesian approach that combines detailed physics simulations with experimental data for the desired statistical inference (Kennedy and O'Hagan 2001; Higdon, Kennedy, Cavendish, Cafeo and Ryne 2004).

The approach given here allows for:

• uncertainty regarding model inputs (i.e. calibration);

• accounting for uncertainty due to limitations on the number of simulations that can be carried out;

• discrepancy between the simulation code and the actual physical system;

• and uncertainty in the observation process that yields the actual field data on the true physical system.

A key problem in statistics and machine learning is inferring suitable structure of a model given some observed data. A Bayesian approach to model comparison makes use of the marginal likelihood of each candidate model to form a posterior distribution over models; unfortunately for most models of interest, notably those containing hidden or latent variables, the marginal likelihood is intractable to compute.

We present the variational Bayesian (VB) algorithm for directed graphical models, which optimises a lower bound approximation to the marginal likelihood in a procedure similar to the standard EM algorithm. We show that for a large class of models, which we call conjugate exponential, the VB algorithm is a straightforward generalisation of the EM algorithm that incorporates uncertainty over model parameters. In a thorough case study using a small class of bipartite DAGs containing hidden variables, we compare the accuracy of the VB approximation to existing asymptotic-data approximations such as the Bayesian Information Criterion (BIC) and the Cheeseman-Stutz (CS) criterion, and also to a sampling based gold standard, Annealed Importance Sampling (AIS). We find that the VB algorithm is empirically superior to CS and BIC, and much faster than AIS. Moreover, we prove that a VB approximation can always be constructed in such a way that guarantees it to be more accurate than the CS approximation.

Nested sampling estimates directly how the likelihood function relates to prior mass. The evidence (alternatively the marginal likelihood, marginal density of the data, or the prior predictive) is immediately obtained by summation. It is the prime result of the computation, and is accompanied by an estimate of numerical uncertainty. Samples from the posterior distribution are an optional by-product, obtainable for any temperature. The method relies on sampling within a hard constraint on likelihood value, as opposed to the softened likelihood of annealing methods. Progress depends only on the shape of the "nested" contours of likelihood, and not on the likelihood values. This invariance (over monotonic re-labelling) allows the method to deal with a class of phase-change problems which effectively defeat thermal annealing.

We introduce a new skew-probit link for item response theory (IRT) by considering an accumulated skew-normal distribution. The model extends the symmetric probit-normal IRT model by considering a new item (or skewness) parameter for the item characteristic curve. A special interpretation is given for this parameter, and a latent linear structure is indicated for the model when an augmented likelihood is considered. Bayesian MCMC inference approach is developed and an efficiency study in the estimation of the model parameters is undertaken for a data set from (Tanner 1996, pg. 190) by using the notion of effective sample size (ESS) as defined in Kass et al. (1999) and the sample size per second (ESS/s) as considered in Sahu (2002). The methodology is illustrated using a data set corresponding to a Mathematical Test applied in Peruvian schools for which a sensitivity analysis of the chosen priors is conducted and also a comparison with seven parametric IRT models is conducted. The main conclusion is that the skew-probit item response model seems to provide the best fit.

Inference proceeds from ingredients chosen by the analyst and data. To validate any inferences drawn it is essential that the inputs chosen be deemed appropriate for the data. In the Bayesian context these inputs consist of both the sampling model and the prior. There are thus two possibilities for failure: the data may not have arisen from the sampling model, or the prior may place most of its mass on parameter values that are not feasible in light of the data (referred to here as prior-data conflict). Failure of the sampling model can only be fixed by modifying the model, while prior-data conflict can be overcome if sufficient data is available. We examine how to assess whether or not a prior-data conflict exists, and how to assess when its effects can be ignored for inferences. The concept of prior-data conflict is seen to lead to a partial characterization of what is meant by a noninformative prior or a noninformative sequence of priors.

Performance evaluations of health services providers burgeons. Similarly, analyzing spatially related health information, ranking teachers and schools, and identification of differentially expressed genes are increasing in prevalence and importance. Goals include valid and efficient ranking of units for profiling and league tables, identification of excellent and poor performers, the most differentially expressed genes, and determining "exceedances" (how many and which unit-specific true parameters exceed a threshold). These data and inferential goals require a hierarchical, Bayesian model that accounts for nesting relations and identifies both population values and random effects for unit-specific parameters. Furthermore, the Bayesian approach coupled with optimizing a loss function provides a framework for computing non-standard inferences such as ranks and histograms.

Estimated ranks that minimize Squared Error Loss (SEL) between the true and estimated ranks have been investigated. The posterior mean ranks minimize SEL and are "general purpose," relevant to a broad spectrum of ranking goals. However, other loss functions and optimizing ranks that are tuned to application-specific goals require identification and evaluation. For example, when the goal is to identify the relatively good (e.g., in the upper 10%) or relatively poor performers, a loss function that penalizes classification errors produces estimates that minimize the error rate. We construct loss functions that address this and other goals, developing a unified framework that facilitates generating candidate estimates, comparing approaches and producing data analytic performance summaries. We compare performance for a fully parametric, hierarchical model with Gaussian sampling distribution under Gaussian and a mixture of Gaussians prior distributions. We illustrate approaches via analysis of standardized mortality ratio data from the United States Renal Data System.

Results show that SEL-optimal ranks perform well over a broad class of loss functions but can be improved upon when classifying units above or below a percentile cut-point. Importantly, even optimal rank estimates can perform poorly in many real-world settings; therefore, data-analytic performance summaries should always be reported.

We introduce a class of multi-scale models for time series. The novel framework couples standard linear models at different levels of resolution via stochastic links across scales. Jeffrey's rule of conditioning is used to revise the implied distributions and ensure that the probability distributions at the different levels are strictly compatible. This results in a new class of models for time series with three key characteristics: this class exhibits a variety of autocorrelation structures based on a parsimonious parameterization, it has the ability to combine information across levels of resolution, and it also has the capacity to emulate long memory processes. The potential applications of such multi-scale models include problems in which it is of interest to develop consistent stochastic models across time-scales and levels of resolution, in order to coherently combine and integrate information arising at different levels of resolution. Bayesian estimation based on MCMC analysis and forecasting based on simulation are developed. One application to the analysis of the flow of a river illustrates the new class of models and its utility.

To measure the impact of Bayesian reasoning, this paper investigates the occurrence of two words, "Bayes" and "Bayesian," since 1970 in journal articles in a variety of disciplines, with a focus on economics and statistics. The growth in statistics is documented, but the growth in economics is largely confined to economic theory/mathematical economics rather than econometrics.