Scientific Background: In developing countries, higher infant mortality is partially caused by poor maternal and fetal nutrition. Clinical trials of micronutrient supplementation are aimed at reducing the risk of infant mortality by increasing birth weight. Because infant mortality is greatest among the low birth weight infants (LBW) ($\leq$ 2500 grams), an effective intervention may need to increase birth weight among the smallest babies. Although it has been demonstrated that supplementation increases the birth weight in a trial conducted in Nepal, there is inconclusive evidence that the supplementation improves their survival. It has been hypothesized that a potential benefit of the treatment on survival among the LBW infants is partly compensated by a null or even harmful effect among the largest infants. Exploratory analyses have suggested that the treatment effect on birth weight might vary with respect to the percentiles of the birth weight distribution.

Data: The methods in this paper are motivated by a double-blind randomized community trial in rural Nepal (Christian et al 2003a,b). The investigators implemented an intervention program to evaluate benefits of the following micronutrient supplementations: folic acid and vitamin A (F+A); folic acid, iron, and vitamin A (F+I+A); folic acid, iron, zinc, and vitamin A (F+I+Z+A); multiple nutrients and vitamin A (M+A). Each micronutrient supplement was administered daily to 1000 pregnant women, who ultimately delivered approximately 800 live-born infants. The team measured the birth weight within 72 hours of delivery and then followed the infants for one year to determine whether or not they survived. In addition, they measured several characteristics of the mother (maternal age, maternal height, arm circumference) and of the infant (weight, length, head and chest circumference).

In this case study we focus on the supplementations F+I+A and M+A as compared to vitamin A only and we address the following scientific questions:

• Is there an overall effect of the treatments on birth weight? Does this effect vary with the percentiles of the birth weight distribution? In particular, is it largest among the LBW infants?

• Is there an overall effect of the treatments on survival? Does this effect vary with the percentiles of the birth weight distribution? In particular, is it largest among the LBW infants?

• Do these percentile-specific effects on birth weight and survival differ by micronutrients?

Statistical Approach: The data analysis is challenged by measurement error and informative missing data in birth weight and survival. In community-based interventions in developing countries, most births occur in the home without assistance from trained birth attendants. Approximately 88% of the babies are measured within 72 hours of the delivery. The remaining 12% are measured between 72 and 2000 hours from the delivery approximately. Hence, weights are obtained at varying times following birth and therefore are imprecise measures of the" true weight at birth". In addition, a high proportion of deaths of young infants occur in the first few hours after birth. If there is a delay in reaching the mother and infant, then many of these infants would not be weighed because they have already died. For example in the F+I+A group, approximately 7% of the birth weight measurements are missing and among this 7%, approximately 34% of the babies have died within 24 hours of the delivery. These babies are likely to have been of lower birth weight than those who survived to be weighed, and therefore, these missing birth weights due to death are likely to be informative.

In this paper we develop a measurement error model with counterfactual variables that address the scientific questions for this birth weight-mortality case study. Our approach integrates Bayesian methods and data augmentation (Tanner and Wong 1987; Tanner 1991; Albert and Chib 1993; Chib and Greenberg 1998) with a counterfactual model and principal stratification (Rubin 1978; Holl 1986; Frangakis and Rubin 2002). We calculate marginal posterior distributions of the treatment effects on birth weight and infant mortality that are allowed to vary with the percentiles of the birth weight distributions. We compare our posterior inferences with two simpler approaches. The first still relies on a Bayesian approach but ignores the uncertainty in the imputation and prediction of the birth weight and does account for the mother's covariates. The second is a simpler re-sampling approach that imputes the missing birth weights (Rubin 1987).

Results and Public Health Impact: First we found that both F+I+A and M+A increase birth weight. However, the F+I+A increases birth weight mainly among the LBW infants, whereas M+A increases birth weight across the entire birth weight distribution compared to vitamin A only. The F+I+A reduces the risk of infant mortality, whereas the M+A slightly increases the risk of early infant mortality, especially among the larger infants.

Currently, recommendations exist to supplement pregnant women in developing countries. This case study provides critical information toward the evaluation and planning of these public health interventions.

Comparing the means of two normal populations is an old problem in mathematical statistics, but there is still no consensus about its most appropriate solution. In this paper we treat the problem of comparing two normal means as a Bayesian decision problem with only two alternatives: either to accept the hypothesis that the two means are equal, or to conclude that the observed data are, under the assumed model, incompatible with that hypothesis. The combined use of an information-theory based loss function, the intrinsic discrepancy (Bernardo and Rueda 2002}, and an objective prior function, the reference prior \citep{Bernardo 1979; Berger and Bernardo 1992), produces a new solution to this old problem which has the invariance properties one should presumably require.

The one way random effects model is analyzed from the Bayesian model selection perspective. From this point of view Bayes factors are the key tool to choose between two models. In order to produce objective Bayes factors objective priors should be assigned to each model.

However, these priors are usually improper provoking a calibration problem which precludes the comparison of the models. To solve this problem several derivations of automatically calibrated objective priors have been proposed among which we quote the intrinsic priors introduced in Berger and Pericchi (1996) and the integral priors introduced in Cano et al. (2006). Here, we focus on the use of integral priors which take advantage of MCMC techniques to produce most of the times unique Bayes factors. Some illustrations are provided.

This paper introduces a novel class of Bayesian models for multivariate time series analysis based on a synthesis of dynamic linear models and graphical models. The synthesis uses sparse graphical modelling ideas to introduce structured, conditional independence relationships in the time-varying, cross-sectional covariance matrices of multiple time series. We define this new class of models and their theoretical structure involving novel matrix-normal/hyper-inverse Wishart distributions. We then describe the resulting Bayesian methodology and computational strategies for model fitting and prediction. This includes novel stochastic evolution theory for time-varying, structured variance matrices, and the full sequential and conjugate updating, filtering and forecasting analysis. The models are then applied in the context of financial time series for predictive portfolio analysis. The improvements defined in optimal Bayesian decision analysis in this example context vividly illustrate the practical benefits of the parsimony induced via appropriate graphical model structuring in multivariate dynamic modelling. We discuss theoretical and empirical aspects of the conditional independence structures in such models, issues of model uncertainty and search, and the relevance of this new framework as a key step towards scaling multivariate dynamic Bayesian modelling methodology to time series of increasing dimension and complexity.

One of the perceived strengths of Bayesian modelling is the ability to include prior information. Although objective or noninformative priors might be preferred in some situations, in many other applications the Bayesian framework offers a real opportunity to formally combine data with information available from experts. The question addressed in this paper is how to elicit this information in a form suitable for prior modelling. Particular attention is paid to geographic data for which maps might be used to assist in the elicitation. Two case studies are used to illustrate the methodology: estimation of city house prices and prediction of presence of a rare species.

Eliciting priors from expert opinion enjoys more efficiency and reliability by avoiding the statistician's potential subjectivity. Since elicitation on the predictive prior probability space requires too-simple priors and may be burdened with additional uncertainties arising from the response model, quantitative elicitation of flexible priors on the direct prior probability space deserves much attention. Motivated by precisely acquiring the shape information for the general location-scale-shape family beyond the limited and simple location-scale family, we investigate multiple numerical procedures for a broad class of priors, as well as interactive graphical protocols for more complicated priors. We highlight the quantile based approaches from several aspects, where Taylor's expansion is demonstrated to be an efficient approximate alternative to work on the regions in which the shape parameter is highly sensitive. By observing inherent associations between the scale and shape parameters, we put more weight on practical solutions under a proper sensitivity index (SI) rather than presumability. Our proposed methodology is demonstrated through skew-normal and Gamma hyper-parameter elicitation where the shape parameter is numerically solved in a stable way. The performance comparisons among different elicitation approaches are also provided.

Setting aside experimental costs, the choice of an experiment is usually formulated in terms of the maximization of a measure of information, often presented as an optimality design criterion. However, there does not seem to be a universal agreement on what objects can qualify as a valid measure of the information in an experiment. In this article we explicitly state a minimal set of requirements that must be satisfied by all such measures. Under that framework, the measure of the information in an experiment is equivalent to the measure of the variability of its likelihood ratio statistics or which is the same, it is equivalent to the measure of the variability of its posterior to prior ratio statistics and to the measure of the variability of the distribution of the posterior distributions yielded by it. The larger that variability, the more peaked the likelihood functions and posterior distributions that tend to be yielded by the experiment, and the more informative the experiment is. By going through various measures of variability, this paper uncovers the unifying link underlying well known information measures as well as information measures that are not yet recognized as such.

The measure of the information in an experiment is then related to the measure of the information in a given observation from it. In this framework, the choice of experiment based on statistical merit only, is posed as a decision problem where the reward is a likelihood ratio or posterior distribution, the utility function is convex, the utility of the reward is the information observed, and the expected utility is the information in an experiment. Finally, the information in an experiment is linked to the information and to the uncertainty in a probability distribution, and we find that the measure of the information in an experiment is not always interpretable as the uncertainty in the prior minus the expected uncertainty in the posterior.

Nonparametric methods for density estimation are examined here. Within a Bayesian setting the construction of an absolutely continuous random probability measure is often required for nonparametric statistical analysis. To achieve this we propose a "partial convexification" procedure of a process, such as the Dirichlet, resulting in a multimodal distribution function with a finite expected number of modes. In agreement with convexity theory results, it is shown that the derived random probability measure admits a density with respect to Lebesgue measure.

We introduce a new method for building classification models when we have prior knowledge of how the classes can be arranged in a hierarchy, based on how easily they can be distinguished. The new method uses a Bayesian form of the multinomial logit (MNL, a.k.a. "softmax") model, with a prior that introduces correlations between the parameters for classes that are nearby in the tree. We compare the performance on simulated data of the new method, the ordinary MNL model, and a model that uses the hierarchy in a different way. We also test the new method on page layout analysis and document classification problems, and find that it performs better than the other methods.