We describe a method for quantifying the lack of fit in a proposed family of distributions. The method involves estimating the posterior distribution of the Kullback-Leibler information between the true distribution generating the data and the proposed family. We include an implementation for discrete data involving Dirichlet Processes, for continuous data involving Dirichlet Process Mixtures, and for regression data involving a common "perturbation" distribution also estimated by a Dirichlet Process Mixture. We examine the effectiveness of the method through simulation. We also show that, for independent, identically distributed discrete data, the posterior distribution from a Dirichlet Process provides a consistent estimate of the KL information. Because the entire posterior distribution is computed, one can readily acquire interval estimates of the distance without resorting to asymptotics.

Multivariate data summarized over areal units (counties, zip codes, etc.) are common in the field of public health. Estimation or testing of geographic boundaries for such data may have varied goals. For example, for data on multiple disease outcomes, we may be interested in a single set of "composite" boundaries for all diseases, separate boundaries for each disease, or both. Different areal wombling (boundary analysis) techniques are needed to meet these different requirements. But in any case, the underlying statistical model needs to account for correlations across both diseases and locations. Utilizing recent developments in multivariate conditionally autoregressive (MCAR) distributions and spatial structural equation modeling, we suggest a variety of Bayesian hierarchical models for multivariate areal boundary analysis, including some that incorporate random neighborhood structure. Many of our models can be implemented via standard software, namely WinBUGS for posterior sampling and $R$ for summarization and plotting. We illustrate our methods using Minnesota county-level esophagus, larynx, and lung cancer data, comparing models that account for both, only one, or neither of the aforementioned correlations. We identify both composite and cancer-specific boundaries, selecting the best statistical model using the DIC criterion. Our results indicate primary boundaries in both the composite and cancer-specific response surface separating the mining- and tourism-oriented northeast counties from the remainder of the state, as well as secondary (residual) boundaries in the Twin Cities metro area.

This paper proposes the construction of a Bayesian specification test based on the encompassing principle for the case of partial observability of latent variables. A structural parametric model (null model) is compared against a nonparametric alternative (alternative model) at the level of latent variables. The null extended model is obtained by incorporating the non Euclidean parameter of the alternative model. This extension is defined through a Bayesian Pseudo-True Value, that makes the null model a reduction by sufficiency of the extended model. The same observability process is introduced in both the null and the alternative models; after integrating out the latent variables, a null and alternative statistical models are accordingly obtained. The comparison is made between the posterior measures of the non Euclidean parameter (of the alternative model) in the extended and in the alternative statistical models. The general development is illustrated with an example where only a linear combination of a latent vector is observed; in the example, the partial observability is known up to the vector defining the observed linear combination. Some identifiability issues are treated and the example shows the operationality and some pitfalls of the proposed test, through a numerical experiment.

Any unperturbed and perturbed posterior density can formally be linked by a mixture. Many divergences between the unperturbed and perturbed posterior density - global measures of influence of the perturbation - are then essentially determined by the Fisher information with respect to the mixing parameter evaluated at the unperturbed density. It is investigated which aspect of change this Fisher information - commonly interpreted as local measure of influence - captures in assessing influence of the perturbation. Under multiplicative modes of perturbation it is nicely interpretable as unperturbed posterior variance of the (log-)perturbation function

We present a new methodology for analysing forensic identification problems involving DNA mixture traces where several individuals may have contributed to the trace. The model used for identification and separation of DNA mixtures is based on a gamma distribution for peak area values. In this paper we illustrate the gamma model and apply it on several real examples from forensic casework.

In this paper, we introduce a Bayesian extended regression model with two-stage priors when the covariate is positive and measured with error. Connections are made with some results in Arellano-Valle and Azzalini (2006), related to the multivariate skew-normal distributions. The usefulness of the proposed model with errors in variables, via the two-stage priors formulated by O'Hagan and Leonard (1976), is illustrated with an example abstracted from Fuller (1987, pg. 18). The main advantage of this extended Bayesian approach is the use of skewed priors, typically rare in most Bayesian applications, and to treat the true value of the explanatory variable as positive, consideration that is sometimes ignored in measurement error models. Such consideration makes naturally the model identifiable, a problem that significantly has troubled users of other approaches listed in the literature. This constraint implies also a strong asymmetry in the distribution of the response variable. Strong connections are shown with results in Li (1997) on non-random samples and with Berkson models, which are important in practical applications. Extensions of Copas and Li's results for models with vector explanatory variables are presented.

In models with conditionally independent observations, it is shown that the posterior variance of the log-likelihood from observation $i$ is a measure of that observation's local influence. This result is obtained by considering the Kullback-Leibler divergence between baseline and case-weight perturbed posteriors, with local influence being the curvature of this divergence evaluated at the baseline posterior. Case-weighting is formulated using quasi-likelihood and hence for binomial or Poisson observations, the posterior variance of an observation's log-likelihood provides a measure of sensitivity to mild mis-specification of its dispersion. In general, the case-weighted posteriors are quasi-posteriors because they do not arise from a formal sampling model. Their propriety is established under a simple sufficient condition. A second local measure of posterior change, the curvature of the Kullback-Leibler divergence between predictive densities, is seen to be the posterior variance (over future observations) of the expected log-likelihood, and can easily be estimated using importance sampling. Suggestions for identifying locally influential observations are given. The methodology is applied to a well known simple linear model dataset, to a nonlinear state-space model, and to a random-effects binary response model.

This paper presents a methodology for cross-validation in the context of Bayesian modelling of situations we loosely refer to as 'inverse problems'. It is motivated by an example from palaeoclimatology in which scientists reconstruct past climates from fossils in lake sediment. The inverse problem is to build a model with which to make statements about climate, given sediment. One natural aspect of this is to examine model fit via 'inverse' cross-validation. We discuss the advantages of inverse cross-validation in Bayesian model assessment. In high-dimensional MCMC studies the inverse cross-validation exercise can be computationally burdensome. We propose a fast method involving very many low-dimensional MCMC runs, using Importance Re-sampling to reduce the dimensionality. We demonstrate that, in addition, the method is particularly suitable for exploring multimodal distributions. We illustrate our proposed methodology with simulation studies and the complex, high-dimensional, motivating palaeoclimate problem.

A variety of simulation-based techniques have been proposed for detection of divergent behaviour at each level of a hierarchical model. We investigate a diagnostic test based on measuring the conflict between two independent sources of evidence regarding a parameter: that arising from its predictive prior given the remainder of the data, and that arising from its likelihood. This test gives rise to a $p$-value that exactly matches or closely approximates a cross-validatory predictive comparison, and yet is more widely applicable. Its properties are explored for normal hierarchical models and in an application in which divergent surgical mortality was suspected. Since full cross-validation is so computationally demanding, we examine full-data approximations which are shown to have only moderate conservatism in normal models. A second example concerns criticism of a complex growth curve model at both observation and parameter levels, and illustrates the issue of dealing with multiple $p$-values within a Bayesian framework. We conclude with the proposal of an overall strategy to detecting divergent behaviour in hierarchical models.