The inferential problem of associating data to mixture components is difficult when components are nearby or overlapping. We introduce a new split-merge Markov chain Monte Carlo technique that efficiently classifies observations by splitting and merging mixture components of a nonconjugate Dirichlet process mixture model. Our method, which is a Metropolis-Hastings procedure with split-merge proposals, samples clusters of observations simultaneously rather than incrementally assigning observations to mixture components. Split-merge moves are produced by exploiting properties of a restricted Gibbs sampling scan. A simulation study compares the new split-merge technique to a nonconjugate version of Gibbs sampling and an incremental Metropolis-Hastings technique. The results demonstrate the improved performance of the new sampler.

The problem of inferring the population structure, linkage disequilibrium pattern, and chromosomal recombination hotspots from genetic polymorphism data is essential for understanding the origin and characteristics of genome variations, with important applications to the genetic analysis of disease propensities and other complex traits. Statistical genetic methodologies developed so far mostly address these problems separately using specialized models ranging from coalescence and admixture models for population structures, to hidden Markov models and renewal processes for recombination; but most of these approaches ignore the inherent uncertainty in the genetic complexity (e.g., the number of genetic founders of a population) of the data and the close statistical and biological relationships among objects studied in these problems. We present a new statistical framework called hidden Markov Dirichlet process (HMDP) to jointly model the genetic recombinations among a possibly infinite number of founders and the coalescence-with-mutation events in the resulting genealogies. The HMDP posits that a haplotype of genetic markers is generated by a sequence of recombination events that select an ancestor for each locus from an unbounded set of founders according to a 1st-order Markov transition process. Conjoining this process with a mutation model, our method accommodates both between-lineage recombination and within-lineage sequence variations, and leads to a compact and natural interpretation of the population structure and inheritance process underlying haplotype data. We have developed an efficient sampling algorithm for HMDP based on a two-level nested Pólya urn scheme, and we present experimental results on joint inference of population structure, linkage disequilibrium, and recombination hotspots based on HMDP. On both simulated and real SNP haplotype data, our method performs competitively or significantly better than extant methods in uncovering the recombination hotspots along chromosomal loci; and in addition it also infers the ancestral genetic patterns and offers a highly accurate map of ancestral compositions of modern populations.

Recent developments in Bayesian computing allow accurate estimation of integrals, making advanced Bayesian analysis feasible. However, some problems remain difficult, such as estimating posterior distributions for variance parameters. For models with three or more variances, this paper proposes a simplex parameterization for the variance structure, which has appealing properties and eases the related burden of specifying a reference prior. This parameterization can be profitably used in several multiple-precision models, including crossed random-effect models, many linear mixed models, smoothed ANOVA, and the conditionally autoregressive (CAR) model with two classes of neighbor relations, often useful for spatial data. The simplex parameterization has at least two attractive features. First, it typically leads to simple MCMC algorithms with good mixing properties regardless of the parameterization used to specify the model's reference prior. Thus, a Bayesian analysis can take computational advantage of the simplex parameterization even if its prior was specified using another parameterization. Second, the simplex parameterization suggests a natural reference prior that is proper, invariant under multiplication of the data by a constant, and which appears to reduce the posterior correlation of smoothing parameters with the error precision. We use simulations to compare the simplex parameterization, with its reference prior, to other parameterizations with their reference priors, according to bias and mean-squared error of point estimates and coverage of posterior 95% credible intervals. The results suggest advantages for the simplex approach, particularly when the error precision is small. We offer results in the context of two real data sets from the fields of periodontics and prosthodontics.

When planning and designing a policy intervention and evaluation, it is important to differentiate between (future) policy interventions we want to evaluate, $F_{T}$, affecting "the world," and experimental allocations, $A_{T}$, affecting "our picture of the world." The policy maker usually has to define a strategy that involves policy assignment and recording mechanisms that will affect the (conditional independence) structure of the data available. Causal inference is sensitive to the specification of these mechanisms. Influence diagrams have been used for causal reasoning within a Bayesian decision-theoretic framework that introduces interventions as decision nodes (Dawid 2002). Design Networks expand this framework by including experimental design decision nodes (Madrigal and Smith 2004). They provide semantics to discuss how a design decision strategy (such as a cluster randomised study) might assist the identification of intervention causal effects. The Design Network framework is extended to Cluster Allocation. It is used to assess identifiability when the experimental unit's level is different from the analysis unit's level, and to discuss the evaluation of cluster- and individual-level future policies. Cases of 'pure' cluster (all individuals in a cluster receiving the same intervention) and 'non-pure' cluster (only a subset receiving the policy) are discussed in terms of causal effects. The representation and analysis of a simplified version of a Mexican social policy programme to alleviate poverty (Progresa) is performed as an illustration of the use of Bayesian hierarchical models to make causal inferences relating to household and community level interventions.

The multiresolution estimator, developed originally in engineering applications as a wavelet-based method for density estimation, has been recently extended and adapted for estimation of hazard functions (Bouman et al. 2005, 2007). Using the multiresolution hazard (MRH) estimator in the Bayesian framework, we are able to incorporate any a priori desired shape and amount of smoothness in the hazard function. The MRH method's main appeal is in its relatively simple estimation and inference procedures, making it possible to obtain simultaneous confidence bands on the hazard function over the entire time span of interest. Moreover, these confidence bands properly reflect the multiple sources of uncertainty, such as multiple centers or heterogeneity in the patient population. Also, rather than the commonly employed approach of estimating covariate effects and the hazard function separately, the Bayesian MRH method estimates all of these parameters jointly, thus resulting in properly adjusted inference about any of the quantities.

In this paper, we extend the previously proposed MRH methods (Bouman et al. 2005, 2007) into the hierarchical multiresolution hazard setting (HMRH), to accommodate the case of separate hazard rate functions within each of several strata as well as some common covariate effects across all strata while accounting for within-stratum correlation. We apply this method to examine patterns of tumor recurrence after treatment for early stage breast cancer, using data from two large-scale randomized clinical trials that have substantially influenced breast cancer treatment standards. We implement the proposed model to estimate the recurrence hazard and explore how the shape differs between patients grouped by a key tumor characteristic (estrogen receptor status) and treatment types, after adjusting for other important patient characteristics such as age, tumor size and progesterone level. We also comment on whether the hazards exhibit non-monotonic patterns consistent with recent hypotheses suggesting multiple hazard change-points at specific time landmarks.

Scientific hypotheses of interest often involve variables that are not available in a single survey. This is a common problem for researchers working with survey data. We propose a model-based approach to provide information about the missing variable. We use a spatial extension of the BART (Bayesian additive regression tree) model. The imputation of the missing variables and inference about the relationship between two variables are obtained simultaneously as posterior inference under the proposed model. The uncertainty due to imputation is automatically accounted for. A simulation analysis and an application to data on self-perceived health status and income are presented.