The precision parameter $\alpha$ plays an important role in the Dirichlet Pro- cess. When assigning a Dirichlet Process prior to the set of probability measures on $\mathbb{R}^k, k \gt 1$, this can be restrictive in the sense that the variability is determined by a single parameter. The aim of this paper is to construct an enrichment foof the Dirichlet Process that is more flexible with respect to the precision parameter yet still conjugate, starting from the notion of enriched conjugate priors, which have been proposed to address an analogous lack of flexibility of standard conjugate priors in a parametric setting. The resulting enriched conjugate prior allows more flexibility in modelling uncertainty on the marginal and conditionals. We describe an enriched urn scheme which characterizes this process and show that it can also be obtained from the stick-breaking representation of the marginal and conditionals. For non atomic base measures, this allows global clustering of the marginal variables and local clustering of the conditional variables. Finally, we consider an application to mixture models that allows for uncertainty between homoskedasticity and heteroskedasticity.

We develop an extension of the classical Zellner's $g$-prior to generalized linear models. Any continuous proper hyperprior $f(g)$ can be used, giving rise to a large class of hyper-$g$ priors. Connections with the literature are described in detail. A fast and accurate integrated Laplace approximation of the marginal likelihood makes inference in large model spaces feasible. For posterior parameter estimation we propose an efficient and tuning-free Metropolis-Hastings sampler. The methodology is illustrated with variable selection and automatic covariate transformation in the Pima Indians diabetes data set.

We address the statistical problem of evaluating $R = P(X \lt Y)$, where $X$ and $Y$ are two independent random variables. Bayesian parametric inference is based on the marginal posterior density of $R$ and has been widely discussed under various distributional assumptions on $X$ and $Y$. This classical approach requires both elicitation of a prior on the complete parameter and numerical integration in order to derive the marginal distribution of $R$. In this paper, we discuss and apply recent advances in Bayesian inference based on higher-order asymptotics and on pseudo-likelihoods, and related matching priors, which allow one to perform accurate inference on the parameter of interest $R$ only, even for small sample sizes. The proposed approach has the advantages of avoiding the elicitation on the nuisance parameters and the computation of multidimensional integrals. From a theoretical point of view, we show that the used prior is a strong matching prior. From an applied point of view, the accuracy of the proposed methodology is illustrated both by numerical studies and by real-life data concerning clinical studies.

We propose the calibrated posterior predictive $p$-value ($cppp$) as an interpretable goodness-of-fit (GOF) measure for regression models in sequential regression multiple imputation (SRMI). The $cppp$ is uniformly distributed under the assumed model, while the posterior predictive $p$-value ($ppp$) is not in general and in particular when the percentage of missing data, $pm$, increases. Uniformity of $cppp$ allows the analyst to evaluate properly the evidence against the assumed model. We show the advantages of $cppp$ over $ppp$ in terms of power in detecting common departures from the assumed model and, more importantly, in terms of robustness with respect to $pm$. In the imputation phase, which provides a complete database for general statistical analyses, default and improper priors are usually needed, whereas the $cppp$ requires a proper prior on regression parameters. We avoid this problem by introducing the use of a minimum training sample that turns the improper prior into a proper distribution. The dependency on the training sample is naturally accounted for by changing the training sample at each step of the SRMI. Our results come from theoretical considerations together with simulation studies and an application to a real data set of anthropometric measures.

Radiocarbon dating is routinely used in paleoecology to build chronologies of lake and peat sediments, aiming at inferring a model that would relate the sediment depth with its age. We present a new approach for chronology building (called "Bacon") that has received enthusiastic attention by paleoecologists. Our methodology is based on controlling core accumulation rates using a gamma autoregressive semiparametric model with an arbitrary number of subdivisions along the sediment. Using prior knowledge about accumulation rates is crucial and informative priors are routinely used. Since many sediment cores are currently analyzed, using different data sets and prior distributions, a robust (adaptive) MCMC is very useful. We use the t-walk (Christen and Fox, 2010), a self adjusting, robust MCMC sampling algorithm, that works acceptably well in many situations. Outliers are also addressed using a recent approach that considers a Student-t model for radiocarbon data. Two examples are presented here, that of a peat core and a core from a lake, and our results are compared with other approaches.

We present a Bayesian surrogate model for the analysis of periodic or quasi-periodic time series data. We describe a computationally efficient implementation that enables Bayesian model comparison. We apply this model to simulated and real exoplanet observations. We discuss the results and demonstrate some of the challenges for applying our surrogate model to realistic exoplanet data sets. In particular, we find that analyses of real world data should pay careful attention to the effects of uneven spacing of observations and the choice of prior for the "jitter" parameter.