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We consider the problem of variable selection in data sets with many response variables and many covariates. A method is proposed that allows some covariates to affect some response variables and not others, and that clusters responses which have similar dependence on the same set of covariates. A Markov chain Monte Carlo procedure is employed to sample from the space of pairwise partitions of covariates and outcomes, where a pair consists of a subset of all outcomes and their associated covariates. We assess the performance of the method on simulated data and apply it to genomic data.
The article by Monni and Tadesse introduces a model for relating large numbers of predictors and responses. That situation typically occurs when investigators are in an exploratory mode. This discussion argues that in such situations the fairly strong assumptions of Monni and Tadesse (e.g., linear regression models with common coefficients for all variables within a cluster of responses) may be counterproductive. If such models are to be used, it is critical that model fit be assessed before relying on their results.
Infectious diseases both within human and animal populations often pose serious health and socioeconomic risks. From a statistical perspective, their prediction is complicated by the fact that no two epidemics are identical due to changing contact habits, mutations of infectious agents, and changing human and animal behaviour in response to the presence of an epidemic. Thus model parameters governing infectious mechanisms will typically be unknown. On the other hand, epidemic control strategies need to be decided rapidly as data accumulate. In this paper we present a fully Bayesian methodology for performing inference and online prediction for epidemics in structured populations. Key features of our approach are the development of an MCMC- (and adaptive MCMC-) based methodology for parameter estimation, epidemic prediction, and online assessment of risk from currently unobserved infections. We illustrate our methods using two complementary studies: an analysis of the 2001 UK Foot and Mouth epidemic, and modelling the potential risk from a possible future Avian Influenza epidemic to the UK Poultry industry.
Based on a constructive representation, which distinguishes between a skewing mechanism $P$ and an underlying symmetric distribution $F$, we introduce two flexible classes of distributions. They are generated by nonparametric modelling of either $P$ or $F$. We examine properties of these distributions and consider how they can help us to identify which aspects of the data are badly captured by simple symmetric distributions. Within a Bayesian framework, we investigate useful prior settings and conduct inference through MCMC methods. On the basis of simulated and real data examples, we make recommendations for the use of our models in practice. Our models perform well in the context of density estimation using the multimodal galaxy data and for regression modelling with data on the body mass index of athletes.
We consider the problem of predicting the achievement of successful pregnancy, in a population of women undergoing treatment for infertility, based on longitudinal measurements of adhesiveness of certain blood lymphocytes. A goal of the analysis is to provide, for each woman, an estimated probability of becoming pregnant. We discuss various existing approaches, including multiple t-tests, mixed models, discriminant analysis and two-stage models. We use a joint model developed by Wange et al. (2000), consisting of a linear mixed effects model for the longitudinal data and a generalized linear model (glm) for the primary endpoint, (here a binary indicator of successful pregnancy). The joint longitudinal/glm model is analogous to the popular joint models for longitudinal and survival data. We estimate the parameters using Bayesian methodology.
Because of the huge number of partitions of even a moderately sized dataset, even when Bayes factors have a closed form, in model-based clustering a comprehensive search for the highest scoring (MAP) partition is usually impossible. However, when each cluster in a partition has a signature and it is known that some signatures are of scientific interest whilst others are not, it is possible, within a Bayesian framework, to develop search algorithms which are guided by these cluster signatures. Such algorithms can be expected to find better partitions more quickly. In this paper we develop a framework within which these ideas can be formalized. We then briefly illustrate the efficacy of the proposed guided search on a microarray time course data set where the clustering objective is to identify clusters of genes with different types of circadian expression profiles.
Contingent valuation models are used in Economics to value non-market goods and can be expressed as binary choice regression models with one of the regression coefficients fixed. A method for flexibly estimating the link function of such binary choice model is proposed by using a Dirichlet process mixture prior on the space of all latent variable distributions, instead of the more restricted distributions in earlier papers. The model is estimated using a novel MCMC sampling scheme that avoids the high autocorrelations in the iterates that usually arise when sampling latent variables that are mixtures. The method allows for variable selection and is illustrated using simulated and real data.
By basing Bayesian probability theory on five axioms, we can give a trivial proof of Cox's Theorem on the product rule and sum rule for conditional plausibility without assuming continuity or differentiablity of plausibility. Instead, we extend the notion of plausibility to apply to unknowns, giving them plausible values. Thus, we combine the best aspects of two approaches to Bayesian probability theory, namely the Cox-Jaynes theory and the de Finetti theory.
We consider Bayes inference for a class of distributions on orientations in 3 dimensions described by $3 \times 3$ rotation matrices. Non-informative priors are identified and Metropolis-Hastings within Gibbs algorithms are used to generate samples from posterior distributions in one-sample and one-way random effects models. A simulation study investigates the performance of Bayes analyses based on non-informative priors in the one-sample case, making comparisons to quasi-likelihood inference. A second simulation study investigates the behavior of posteriors for some informative priors. Bayes one-way random effect analyses of orientation matrix data are then developed and the Bayes methods are illustrated in a materials science application.