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The goal of this paper is to contrast and survey the major advances in two of the most commonly used high-dimensional techniques, namely, the Lasso and horseshoe regularization. Lasso is a gold standard for predictor selection while horseshoe is a state-of-the-art Bayesian estimator for sparse signals. Lasso is fast and scalable and uses convex optimization whilst the horseshoe is nonconvex. Our novel perspective focuses on three aspects: (i) theoretical optimality in high-dimensional inference for the Gaussian sparse model and beyond, (ii) efficiency and scalability of computation and (iii) methodological development and performance.

This paper considers Bayesian regularisation using global–local shrinkage priors in the multivariate general linear model when there are many more explanatory variables than observations. We adopt priors’ structures used extensively in univariate problems (conjugate and non-conjugate with tail behaviour ranging from polynomial to exponential) and consider how the addition of error correlation in the multivariate set-up affects the performance of these priors. Two different datasets (from drug discovery and chemometrics) with many covariates are used for comparison, and these are supplemented by a small simulation study to corroborate the role of error correlation. We find that structural assumptions of the prior distribution on regression coefficients can be more significant than the tail behaviour. In particular, if the structural assumption of conjugacy is used, the performance of the posterior predictive distribution deteriorates relative to non-conjugate choices as the error correlation becomes stronger.

It is now widely recognized that small area estimation (SAE) needs to be model-based. Global-local (GL) shrinkage priors for random effects are important in sparse situations where many areas’ level effects do not have a significant impact on the response beyond what is offered by covariates. We propose in this paper a hierarchical multivariate model with GL priors. We prove the propriety of the posterior density when the regression coefficient matrix has an improper uniform prior. Some concentration inequalities are derived for the tail probabilities of the shrinkage estimators. The proposed method is illustrated via both data analysis and simulations.

Estimating demand for large assortments of differentiated goods requires the specification of a demand system that is sufficiently flexible. However, flexible models are highly parameterized so estimation requires appropriate forms of regularization to avoid overfitting. In this paper, we study the specification of Bayesian shrinkage priors for pairwise product substitution parameters. We use a log-linear demand system as a leading example. Log-linear models are parameterized by own and cross-price elasticities, and the total number of elasticities grows quadratically in the number of goods. Traditional regularized estimators shrink regression coefficients towards zero which can be at odds with many economic properties of price effects. We propose a hierarchical extension of the class of global-local priors commonly used in regression modeling to allow the direction and rate of shrinkage to depend on a product classification tree. We use both simulated data and retail scanner data to show that, in the absence of a strong signal in the data, estimates of price elasticities and demand predictions can be improved by imposing shrinkage to higher-level group elasticities rather than zero.

This study investigates the time-varying effects of international uncertainty shocks. I use a global vector autoregressive model with drifting coefficients and factor stochastic volatility in the errors to model the G7 economies jointly. The measure of uncertainty is constructed by estimating a time-varying scalar driving the innovation variances of the latent factors, which is also included in the conditional mean of the process. To achieve regularization, I use Bayesian techniques for estimation, and rely on hierarchical global–local priors to shrink the high-dimensional multivariate system towards sparsity. I compare the obtained econometric measure of uncertainty to alternative indices and discuss commonalities and differences. Moreover, I find that international uncertainty may differ substantially compared to identically constructed domestic measures. Structural inference points towards pronounced real and financial effects of uncertainty shocks in all considered economies. These effects are subject to heterogeneities over time and the cross-section, providing empirical evidence in favor of using the flexible econometric framework introduced in this study.

Interval datasets are not uncommon in many disciplines including medical experiments, econometric studies, environmental studies etc. For modeling interval data traditionally separate models are used for modeling the center and the radius of the response variable. In this article, we consider a Bayesian regression framework for jointly modeling the center and the radius of the intervals corresponding to the response, and then use appropriate priors for variable selection. Unlike the traditional setting, both the centres and the radii of all the predictors are used for modeling the center and the radius of response. We consider spike and slab priors for the regression coefficients corresponding to the centers (radii) of the predictors while modeling the center (radius) of the response, and global–local shrinkage prior for the coefficients corresponding to the radii (centers) of the predictors. Through extensive simulation studies, we illustrate the effectiveness of our proposed variable selection approach for the analysis and prediction of interval datasets. Finally, we analyze a real dataset from a clinical trial related to the Acute Lymphocytic Leukemia (ALL), and then select the important set of predictors for modeling the lymphocyte count which is an important biomarker for ALL. Our numerical studies show that the proposed approach is efficient, and it provides a powerful statistical inference for handling interval datasets.

We discuss a Bayesian hierarchical copula model for clusters of financial time series. A similar approach has been developed in recent paper. However, the prior distributions proposed there do not always provide a proper posterior. In order to circumvent the problem, we adopt a proper global–local shrinkage prior, which is also able to account for potential dependence structures among different clusters. The performance of the proposed model is presented via simulations and a real data analysis.

From practical point of view, in a two-level hierarchical model, the variance of second-level usually has a tendency to change through sub-populations. The existence of this kind of local (or intrinsic ) heteroscedasticity is a major concern in the application of statistical modeling. The main purpose of this study is to construct a Bayesian methodology via shrinkage priors in order to estimate the interesting parameters under local heteroscedasticity. The suggested methodology for this issue is to use of a class of the local-global shrinkage priors, called Dirichlet-Laplace priors. The optimal posterior concentration and straightforward posterior computation are the appealing properties of these priors. Two real data sets are analyzed to illustrate the proposed methodology.

Penalized regression methods, such as L ₁ regularization, are routinely used in high-dimensional applications, and there is a rich literature on optimality properties under sparsity assumptions. In the Bayesian paradigm, sparsity is routinely induced through two-component mixture priors having a probability mass at zero, but such priors encounter daunting computational problems in high dimensions. This has motivated continuous shrinkage priors, which can be expressed as global-local scale mixtures of Gaussians, facilitating computation. In contrast to the frequentist literature, little is known about the properties of such priors and the convergence and concentration of the corresponding posterior distribution. In this article, we propose a new class of Dirichlet–Laplace priors, which possess optimal posterior concentration and lead to efficient posterior computation. Finite sample performance of Dirichlet–Laplace priors relative to alternatives is assessed in simulated and real data examples.

We provide a framework for assessing the default nature of a prior distribution using the property of regular variation, which we study for global-local shrinkage priors. In particular, we show that the horseshoe priors, originally designed to handle sparsity, are regularly varying and thus are appropriate for default Bayesian analysis. To illustrate our methodology, we discuss four problems of noninformative priors that have been shown to be highly informative for nonlinear functions. In each case, we show that global-local horseshoe priors perform as required. Global-local shrinkage priors can separate a low-dimensional signal from high-dimensional noise even for nonlinear functions.