This paper addresses the use of Jeffreys priors in the context of univariate three-parameter location-scale models, where skewness is introduced by differing scale parameters either side of the location. We focus on various commonly used parameterizations for these models. Jeffreys priors are shown to lead to improper posteriors in the wide and practically relevant class of distributions obtained by skewing scale mixtures of normals. Easily checked conditions under which independence Jeffreys priors can be used for valid inference are derived. We also investigate two alternative priors, one of which is shown to lead to valid Bayesian inference for all practically interesting parameterizations of these models and is our recommendation to practitioners. We illustrate some of these models using real data.

We discuss Rubio and Steel (2014). We discuss whether Jeffreys priors are worth the attention given to them, then move on to discuss the concepts of valid Bayesian inference and benchmark Bayesian inference. We briefly investigate the skew-normal and skew-$t(4)$ models for variables in the Australian Institute of Sport (AIS) data to investigate the range of estimates that occur for the skewness parameter. The discussion closes by wondering whether we shouldn’t just use a Dirichlet Process Mixture instead of a skew-normal or skew-$t$.

Chain Event Graphs (CEGs) are proving to be a useful framework for modelling discrete processes which exhibit strong asymmetric dependence structures between the variables of the problem. In this paper we exploit this framework to represent processes where missingness is influential and data cannot plausibly be hypothesised to be missing at random in all situations. We develop new classes of models where data are missing not at random but nevertheless exhibit context-specific symmetries which are captured by the CEG. We show that it is possible to score each model efficiently and in closed form. Hence standard Bayesian selection methods can be used to search over a wide variety of models, each with its own explanatory narrative. One of the advantages of this method is that the selected maximum a posteriori model and other closely scoring models can be easily read back to the client in a graphically transparent way. The efficacy of our methods are illustrated using a cerebral palsy cohort study, analysing their survival with respect to weight at birth and various disabilities.

We consider sufficiency for Bayes linear revision for multivariate multiple regression problems, and in particular where we have a sequence of multivariate observations at different matrix design points, but with common parameter vector. Such sequences are not usually exchangeable. However, we show that there is a sequence of transformed observations which is exchangeable and we demonstrate that their mean is sufficient both for Bayes linear revision of the parameter vector and for prediction of future observations. We link these ideas to making revisions of belief over replicated structure such as graphical templates of model relationships. We show that the sufficiencies lead to natural residual collections and thence to sequential diagnostic assessments. We show how each finite regression problem corresponds to a parallel implied infinite exchangeable sequence which may be exploited to solve the sample-size design problem. Bayes linear methods are based on limited specifications of belief, usually means, variances, and covariances. As such, the methodology is well suited to high-dimensional regression problems where a full Bayesian analysis is difficult or impossible, but where a linear Bayes approach offers a pragmatic way to combine judgements with data in order to produce posterior summaries.

Differential geometric Markov Chain Monte Carlo (MCMC) strategies exploit the geometry of the target to achieve convergence in fewer MCMC iterations at the cost of increased computing time for each of the iterations. Such computational complexity is regarded as a potential shortcoming of geometric MCMC in practice. This paper suggests that part of the additional computing required by Hamiltonian Monte Carlo and Metropolis adjusted Langevin algorithms produces elements that allow concurrent implementation of the zero variance reduction technique for MCMC estimation. Therefore, zero variance geometric MCMC emerges as an inherently unified sampling scheme, in the sense that variance reduction and geometric exploitation of the parameter space can be performed simultaneously without exceeding the computational requirements posed by the geometric MCMC scheme alone. A MATLAB package is provided, which implements a generic code framework of the combined methodology for a range of models.

A new method for posterior simulation is proposed, based on the combination of higher-order asymptotic results with the inverse transform sampler. This method can be used to approximate marginal posterior distributions, and related quantities, for a scalar parameter of interest, even in the presence of nuisance parameters. Compared to standard Markov chain Monte Carlo methods, its main advantages are that it gives independent samples at a negligible computational cost, and it allows prior sensitivity analyses under the same Monte Carlo variation. The method is illustrated by a genetic linkage model, a normal regression with censored data and a logistic regression model.

In this work we propose a model-based clustering method for time series. The model uses an almost surely discrete Bayesian nonparametric prior to induce clustering of the series. Specifically we propose a general Poisson-Dirichlet process mixture model, which includes the Dirichlet process mixture model as a particular case. The model accounts for typical features present in a time series like trends, seasonal and temporal components. All or only part of these features can be used for clustering according to the user. Posterior inference is obtained via an easy to implement Markov chain Monte Carlo (MCMC) scheme. The best cluster is chosen according to a heterogeneity measure as well as the model selection criterion LPML (logarithm of the pseudo marginal likelihood). We illustrate our approach with a dataset of time series of share prices in the Mexican stock exchange.

In this article we discuss estimation of generalized threshold regression models in settings when the threshold parameter lacks identifiability. In particular, if estimation of the regression coefficients is associated with high uncertainty and/or the difference between regimes is small, estimators of the threshold and, hence, of the whole model can be strongly affected. A new regularized Bayesian estimator for generalized threshold regression models is proposed. We derive conditions for superiority of the new estimator over the standard likelihood one in terms of mean squared error. Simulations confirm excellent finite sample properties of the suggested estimator, especially in the critical settings. The practical relevance of our approach is illustrated by two real-data examples already analyzed in the literature.

In this paper, we construct an objective prior for the degrees of freedom of a $t$ distribution, when the parameter is taken to be discrete. This parameter is typically problematic to estimate and a problem in objective Bayesian inference since improper priors lead to improper posteriors, whilst proper priors may dominate the data likelihood. We find an objective criterion, based on loss functions, instead of trying to define objective probabilities directly. Truncating the prior on the degrees of freedom is necessary, as the $t$ distribution, above a certain number of degrees of freedom, becomes the normal distribution. The defined prior is tested in simulation scenarios, including linear regression with $t$-distributed errors, and on real data: the daily returns of the closing Dow Jones index over a period of 98 days.

In many applications it is of interest to determine a limited number of important explanatory factors (representing groups of potentially overlapping predictors) rather than original predictor variables. The often imposed requirement that the clustered predictors should enter the model simultaneously may be limiting as not all the variables within a group need to be associated with the outcome. Within-group sparsity is often desirable as well. Here we propose a Bayesian variable selection method, which uses the grouping information as a means of introducing more equal competition to enter the model within the groups rather than as a source of strict regularization constraints. This is achieved within the context of Bayesian LASSO (least absolute shrinkage and selection operator) by allowing each regression coefficient to be penalized differentially and by considering an additional regression layer to relate individual penalty parameters to a group identification matrix. The proposed hierarchical model therefore enables inference simultaneously on two levels: (1) the regression layer for the continuous outcome in relation to the predictors and (2) the regression layer for the penalty parameters in relation to the grouping information. Both situations with overlapping and non-overlapping groups are applicable. The method does not assume within-group homogeneity across the regression coefficients, which is implicit in many structured penalized likelihood approaches. The smoothness here is enforced at the penalty level rather than within the regression coefficients. To enhance the potential of the proposed method we develop two rapid computational procedures based on the expectation maximization (EM) algorithm, which offer substantial time savings in applications where the high-dimensionality renders Markov chain Monte Carlo (MCMC) approaches less practical. We demonstrate the usefulness of our method in predicting time to death in glioblastoma patients using pathways of genes.