In randomized experiments with noncompliance, one might wish to focus on compliers rather than on the overall sample. In this vein, Rubin (1998) argued that testing for the complier average causal effect and averaging permutation-based $p$-values over the posterior distribution of the compliance types could increase power as compared to general intent-to-treat tests. The general scheme is a repeated two-step process: impute missing compliance types and conduct a permutation test with the completed data. In this paper, we explore this idea further, comparing the use of discrepancy measures—which depend on unknown but imputed parameters—to classical test statistics and contrasting different approaches for imputing the unknown compliance types. We also examine consequences of model misspecification in the imputation step, and discuss to what extent this additional modeling undercuts the advantage of permutation tests being model independent. We find that, especially for discrepancy measures, modeling choices can impact both power and validity. In particular, imputing missing compliance types under the null can radically reduce power, but not doing so can jeopardize validity. Fortunately, using covariates predictive of compliance type in the imputation can mitigate these results. We also compare this overall approach to Bayesian model-based tests, that is, tests that are directly derived from posterior credible intervals, under both correct and incorrect model specification.

In this article, we present some specific aspects of symmetric Gamma process mixtures for use in regression models. First we propose a new Gibbs sampler for simulating the posterior. The algorithm is tested on two examples, the mean regression problem with normal errors, and the reconstruction of two dimensional CT images. In a second time, we establish posterior rates of convergence related to the mean regression problem with normal errors. For location-scale and location-modulation mixtures the rates are adaptive over Hölder classes, and in the case of location-modulation mixtures are nearly optimal.

The power-expected-posterior (PEP) prior provides an objective, automatic, consistent and parsimonious model selection procedure. At the same time it resolves the conceptual and computational problems due to the use of imaginary data. Namely, (i) it dispenses with the need to select and average across all possible minimal imaginary samples, and (ii) it diminishes the effect that the imaginary data have upon the posterior distribution. These attributes allow for large sample approximations, when needed, in order to reduce the computational burden under more complex models. In this work we generalize the applicability of the PEP methodology, focusing on the framework of generalized linear models (GLMs), by introducing two new PEP definitions which are in effect applicable to any general model setting. Hyper-prior extensions for the power parameter that regulates the contribution of the imaginary data are introduced. We further study the validity of the predictive matching and of the model selection consistency, providing analytical proofs for the former and empirical evidence supporting the latter. For estimation of posterior model and inclusion probabilities we introduce a tuning-free Gibbs-based variable selection sampler. Several simulation scenarios and one real life example are considered in order to evaluate the performance of the proposed methods compared to other commonly used approaches based on mixtures of $g$-priors. Results indicate that the GLM-PEP priors are more effective in the identification of sparse and parsimonious model formulations.

We consider the specification of an informative prior distribution for the probabilities in a multinomial model. We utilise vine copulas: flexible multivariate distributions built using bivariate copulas stacked in a tree structure. We take advantage of a specific vine structure, called a D-vine, to separate the specification of the multivariate prior distribution into that of marginal distributions for the probabilities and parameter values for the bivariate copulas in the vine. We provide guidance on each of the choices to be made in the prior specification and each of the questions to ask the expert to specify the model parameters within the context of an engineering application. We then give full details of the approach for the general problem.

We introduce a Bayesian estimator of the underlying class structure in the stochastic block model, when the number of classes is known. The estimator is the posterior mode corresponding to a Dirichlet prior on the class proportions, a generalized Bernoulli prior on the class labels, and a beta prior on the edge probabilities. We show that this estimator is strongly consistent when the expected degree is at least of order ${log}^{2}n$, where $n$ is the number of nodes in the network.

We propose a new scheme for selecting pool states for the embedded Hidden Markov Model (HMM) Markov Chain Monte Carlo (MCMC) method. This new scheme allows the embedded HMM method to be used for efficient sampling in state space models where the state can be high-dimensional. Previously, embedded HMM methods were only applicable to low-dimensional state-space models. We demonstrate that using our proposed pool state selection scheme, an embedded HMM sampler can have similar performance to a well-tuned sampler that uses a combination of Particle Gibbs with Backward Sampling (PGBS) and Metropolis updates. The scaling to higher dimensions is made possible by selecting pool states locally near the current value of the state sequence. The proposed pool state selection scheme also allows each iteration of the embedded HMM sampler to take time linear in the number of the pool states, as opposed to quadratic as in the original embedded HMM sampler.

The method of Bayesian variable selection via penalized credible regions separates model fitting and variable selection. The idea is to search for the sparsest solution within the joint posterior credible regions. Although the approach was successful, it depended on the use of conjugate normal priors. More recently, improvements in the use of global-local shrinkage priors have been made for high-dimensional Bayesian variable selection. In this paper, we incorporate global-local priors into the credible region selection framework. The Dirichlet–Laplace (DL) prior is adapted to linear regression. Posterior consistency for the normal and DL priors are shown, along with variable selection consistency. We further introduce a new method to tune hyperparameters in prior distributions for linear regression. We propose to choose the hyperparameters to minimize a discrepancy between the induced distribution on R-square and a prespecified target distribution. Prior elicitation on R-square is more natural, particularly when there are a large number of predictor variables in which elicitation on that scale is not feasible. For a normal prior, these hyperparameters are available in closed form to minimize the Kullback–Leibler divergence between the distributions.

Aim of this contribution is to propose a new regression model for continuous variables bounded to the unit interval (e.g. proportions) based on the flexible beta (FB) distribution. The latter is a special mixture of two betas, which greatly extends the shapes of the beta distribution mainly in terms of asymmetry, bimodality and heavy tail behaviour. Its special mixture structure ensures good theoretical properties, such as strong identifiability and likelihood boundedness, quite uncommon for mixture models. Moreover, it makes the model computationally very tractable also within the Bayesian framework here adopted.

At the same time, the FB regression model displays easiness of interpretation as well as remarkable fitting capacity for a variety of data patterns, including unimodal and bimodal ones, heavy tails and presence of outliers. Indeed, simulation studies and applications to real datasets show a general better performance of the FB regression model with respect to competing ones, namely the beta (Ferrari and Cribari-Neto, 2004) and the beta rectangular (Bayes et al., 2012), in terms of precision of estimates, goodness of fit and posterior predictive intervals.

Sampling errors in nested sampling parameter estimation differ from those in Bayesian evidence calculation, but have been little studied in the literature. This paper provides the first explanation of the two main sources of sampling errors in nested sampling parameter estimation, and presents a new diagrammatic representation for the process. We find no current method can accurately measure the parameter estimation errors of a single nested sampling run, and propose a method for doing so using a new algorithm for dividing nested sampling runs. We empirically verify our conclusions and the accuracy of our new method.

Normalized compound random measures are flexible nonparametric priors for related distributions. We consider building general nonparametric regression models using normalized compound random measure mixture models. Posterior inference is made using a novel pseudo-marginal Metropolis-Hastings sampler for normalized compound random measure mixture models. The algorithm makes use of a new general approach to the unbiased estimation of Laplace functionals of compound random measures (which includes completely random measures as a special case). The approach is illustrated on problems of density regression.

Bayesian model averaging is flawed in the $\mathcal{M}$-open setting in which the true data-generating process is not one of the candidate models being fit. We take the idea of stacking from the point estimation literature and generalize to the combination of predictive distributions. We extend the utility function to any proper scoring rule and use Pareto smoothed importance sampling to efficiently compute the required leave-one-out posterior distributions. We compare stacking of predictive distributions to several alternatives: stacking of means, Bayesian model averaging (BMA), Pseudo-BMA, and a variant of Pseudo-BMA that is stabilized using the Bayesian bootstrap. Based on simulations and real-data applications, we recommend stacking of predictive distributions, with bootstrapped-Pseudo-BMA as an approximate alternative when computation cost is an issue.