... In this article we present the general outlook and discuss general families of elaborations for use in practice; the exponential connection elaboration plays a key role. We then describe model elaborations for use in diagnosing: departures from normality, goodness of fit in generalized linear models, and variable selection in regression and outlier detection. We illustrate our approach with two applications.

This paper proposes a new approach to sparsity called the horseshoe estimator. The horseshoe is a close cousin of other widely used Bayes rules arising from, for example, double-exponential and Cauchy priors, in that it is a member of the same family of multivariate scale mixtures of normals. But the horseshoe enjoys a number of advantages over existing approaches, including its robustness, its adaptivity to different sparsity patterns, and its analytical tractability. We prove two theorems that formally characterize both the horseshoe’s adeptness at large outlying signals, and its super-efficient rate of convergence to the correct estimate of the sampling density in sparse situations. Finally, using a combination of real and simulated data, we show that the horseshoe estimator corresponds quite closely to the answers one would get by pursuing a full Bayesian model-averaging approach using a discrete mixture prior to model signals and noise.

The problem of graduating a sequence of data values can be cast as a statistical estimation problem. In particular, the Bayesian approach is at-tractive due to its ability to formally incorporate known ordering and smooth-ness conditions for the graduated values into the estimation structure. However, this approach has not been widely adopted in practice, primarily because of the arduousness of specifying the prior distributions for the graduated values and carrying out the necessary numerical integrations. This paper presents simple Bayesian graduation models that substantially ease the prior elicita-tion burden; it also describes a Monte Carlo integration approach that greatly reduces the computational load. The method is presented in generality and subsequently illustrated with two examples, one from the realm of health insurance and the other from the more traditional graduation context of mortality table construction. It is hoped that the method will stimulate greater use of the Bayesian paradigm within the actuarial community. 1.

A variation of the Gibbs sampling scheme is defined by driving the simulated Markov chain by the conditional distributions of an approximation to the posterior rather than the posterior distribution itself. Choosing a multivariate normal mixture form for the approximation enables reparametrization which is crucial to improve convergence in the Gibbs sampler. Using an approximation to the posterior density also opens the possiblity to include a learning process about the - in the operational sense of evaluating posterior integrals - unknown posterior density in the algorithm. While ideally this should be done using available pointwise evaluations of the posterior density, this is too difficult in a general framework and we use instead the currently available Monte Carlo sample to adjust the approximating density. This is done using a simple multivariate implementation of the mixture of Dirichlet density estimation algorithm. Keywords: Markov chain Monte Carlo, Bayesian sampling, stocha...

We use Lévy processes to generate joint prior distributions for a location parameter β = (β1,..., βp) as p grows large. This approach, which generalizes normal scale-mixture priors to an infinite-dimensional setting, has a number of connections with mathematical finance and Bayesian nonparametrics. We argue that it provides an intuitive framework for generating new regularization penalties and shrinkage rules; for performing asymptotic analysis on existing models; and for simplifying proofs of some classic results on normal scale mixtures.

This paper is a review of computational strategies for Bayesian shrinkage and variable selection in the linear model. Our focus is less on traditional MCMC methods, which are covered in depth by earlier review papers. Instead, we focus more on recent innovations in stochastic search and adaptive MCMC, along with some comparatively new research on shrinkage priors. One of our conclusions is that true MCMC seems inferior to stochastic search if one’s goal is to discover good models, but that stochastic search can result in biased estimates of variable inclusion probabilities. We also find reasons to question the accuracy of inclusion probabilities generated by traditional MCMC on high-dimensional, nonorthogonal problems, though the matter is far from settled. Some key words: adaptive MCMC; linear models; shrinkage priors; stochastic search; variable selection 1

Using data from the National Longitudinal Survey of Youth (NLSY) we introduce and estimate various Bayesian hierarchical models that investigate the nature of unobserved hetero-geneity in returns to schooling. We consider a variety of possible forms for the heterogeneity, some motivated by previous theoretical and empirical work and some new ones, and let the data decide among the competing specifications. Empirical results indicate that heterogeneity is present in re-turns to education. Furthermore, we find strong evidence that the heterogeneity follows a continuous rather than discrete distribution, and that bivariate Normality provides a very reasonable description of individual-level heterogeneity in intercepts and returns to schooling.

In this paper, we apply a Bayesian approach to test for a structural break with unknown breakpoint in an empirical model of excess returns that allows the equity premium to change in response to recurrent changes in the level of volatility. The main questions we seek to answer with our approach are the following: Is there evidence of changes in the equity premium over time? If so, can these changes be explained as the consequence of recurrent changes in the level of volatility? Or, alternatively, does the equity premium undergo a one-time permanent structural break? For monthly excess returns on a value-weighted portfolio of NYSE stocks between 1926-1991, we find strong evidence for a structural break in the Markov-switching variance process around 1941. However, the data provide little evidence of a concurrent structural break in the equity premium. Instead, the data suggest that changes in the equity premium are mainly a consequence of recurrent changes in the level of volatility. Ke...

The authors discuss prior distributions that are conjugate to the multivariate normal likelihood when some of the observations are incomplete. They present a general class of priors for incorporating information about unidentified parameters in the covariance matrix. They analyze the special case of monotone patterns of missing data, providing an explicit recursive form for the posterior distribution resulting from a conjugate prior distribution. They develop an importance sampling and a Gibbs sampling approach to sample from a general posterior distribution and compare the two methods. R ESUM E Les auteurs proposent des lois a priori conjuguees a la vraisemblance normale multivariee en presence d'observations incompletes. Ils decrivent une classe generale de lois a priori permettant d'incorporer de l'information concernant des parametres non identifies dans la matrice de covariance. Ils analysent le cas special oulepatrondesdonnees manquantes est monotone; ils montrent comment calcule...

Bayes factor can be used to choose between two speci ed models, based on available observations, without any requirement of nested models. In the case of equal prior probabilities of the two models, Bayes factor is given as the posterior odds ratio of the models. Evaluation of Bayes factor includes solving of integrals, which in most cases can only be solved numerically. This work presents an McMC algorithm constructed to estimate Bayes factor when the available observations contain sampling errors. The McMC algorithm allows simultaneous sampling of the model parameters.