Graphical model learning and inference are often performed using Bayesian techniques. In particular, learning is usually performed in two separate steps. First, the graph structure is learned from the data; then the parameters of the model are estimated conditional on that graph structure. While the probability distributions involved in this second step have been studied in depth, the ones used in the first step have not been explored in as much detail.

In this paper, we will study the prior and posterior distributions defined over the space of the graph structures for the purpose of learning the structure of a graphical model. In particular, we will provide a characterisation of the behaviour of those distributions as a function of the possible edges of the graph. We will then use the properties resulting from this characterisation to define measures of structural variability for both Bayesian and Markov networks, and we will point out some of their possible applications.

In this paper, similar to the frequentist asymptotic theory, we present large sample theory for the normalized inverse-Gaussian process and its corresponding quantile process. In particular, when the concentration parameter is large, we establish the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem for the normalized inverse-Gaussian process and its related quantile process. We also derive a finite sum representation that converges almost surely to the Ferguson and Klass representation of the normalized inverse-Gaussian process. This almost sure approximation can be used to simulate the normalized inverse-Gaussian process.

We discuss the definition of a Bayes factor and develop some inequalities relevant to Bayesian inferences. An approach to hypothesis assessment based on the computation of a Bayes factor, a measure of the strength of the evidence given by the Bayes factor via a posterior probability, and the point where the Bayes factor is maximized is recommended. It is also recommended that the a priori properties of a Bayes factor be considered to assess possible bias inherent in the Bayes factor. This methodology can be seen to deal with many of the issues and controversies associated with hypothesis assessment. We present an application to a two-way analysis.

Demographic estimates for small areas within a country have many uses. Subnational population estimation is, however, difficult, requiring the synthesis of multiple inconsistent datasets. Current methods have important limitations, including a heavy reliance on ad hoc adjustment and limited allowance for uncertainty. In this paper we demonstrate how subnational population estimation can be carried out within a formal Bayesian framework. The core of the framework is a demographic account, providing a complete description of the demographic system. Regularities within the demographic account are described by a system model. The relationship between the demographic account and the observable data is described by an observation model. Posterior simulation is carried out using Markov chain Monte Carlo methods. We illustrate the methods using data for six regions within New Zealand.

The receiver operating characteristic (ROC) curve is the most widely used measure for evaluating the discriminatory performance of a continuous biomarker. Incorporating covariates in the analysis can potentially enhance information gathered from the biomarker, as its discriminatory ability may depend on these. In this paper we propose a dependent Bayesian nonparametric model for conditional ROC estimation. Our model is based on dependent Dirichlet processes, where the covariate-dependent ROC curves are indirectly modeled using probability models for related probability distributions in the diseased and healthy groups. Our approach allows for the entire distribution in each group to change as a function of the covariates, provides exact posterior inference up to a Monte Carlo error, and can easily accommodate multiple continuous and categorical predictors. Simulation results suggest that, regarding the mean squared error, our approach performs better than its competitors for small sample sizes and nonlinear scenarios. The proposed model is applied to data concerning diagnosis of diabetes.

We define a new Bayesian predictor called the posterior weighted median (PWM) and compare its performance to several other predictors including the Bayes model average under squared error loss, the Barbieri-Berger median model predictor, the stacking predictor, and the model average predictor based on Akaike’s information criterion. We argue that PWM generally gives better performance than other predictors over a range of $\mathcal{M}$-complete problems. This range is between the $\mathcal{M}$-closed-$\mathcal{M}$-complete boundary and the $\mathcal{M}$-complete-$\mathcal{M}$-open boundary. Indeed, as a problem gets closer to $\mathcal{M}$-open, it seems that $\mathcal{M}$-complete predictive methods begin to break down. Our comparisons rest on extensive simulations and real data examples.

As a separate issue, we introduce the concepts of the ‘Bail out effect’ and the ‘Bail in effect’. These occur when a predictor gives not just poor results but defaults to the simplest model (‘bails out’) or to the most complex model (‘bails in’) on the model list. Either can occur in $\mathcal{M}$-complete problems when the complexity of the data generator is too high for the predictor scheme to accommodate.

A wide range of methods, Bayesian and others, tackle regression when there are many variables. In the Bayesian context, the prior is constructed to reflect ideas of variable selection and to encourage appropriate shrinkage. The prior needs to be reasonably robust to different signal to noise structures. Two simple evergreen prior constructions stem from ridge regression on the one hand and g-priors on the other. We seek to embed recent ideas about sparsity of the regression coefficients and robustness into these priors. We also explore the gains that can be expected from these differing approaches.

We propose a Bayesian adaptive dose-finding design for drug combination trials with delayed toxicity. We model the dose-toxicity relationship using the Finney model, a model widely used in drug-drug interaction studies. The intuitive interpretations of the Finney model facilitate incorporating the available prior dose-toxicity information from single-agent trials into combination trials through prior elicitation. We treat unobserved delayed toxicity outcomes as missing data and handle them using Bayesian data augmentation. We conduct extensive simulation studies to examine the operating characteristics of the proposed method under various practical scenarios. Results show that the proposed design is safe and able to select the target dose combinations with high probabilities.

We introduce an autoregressive model for responses that are restricted to lie on the unit interval, with beta-distributed marginals. The model includes strict stationarity as a special case, and is based on the introduction of a series of latent random variables with a simple hierarchical specification that achieves the desired dependence while being amenable to posterior simulation schemes. We discuss the construction, study some of the main properties, and compare it with alternative models using simulated data. We finally illustrate the usage of our proposal by modelling a yearly series of unemployment rates.