Triple the Gamma—A Unifying Shrinkage Prior for Variance and Variable Selection in Sparse State Space and TVP Models

Generalized cumulative shrinkage process priors with applications to sparse Bayesian factor analysis

The paper discusses shrinkage priors which impose increasing shrinkage in a sequence of parameters. We review the cumulative shrinkage process (CUSP) prior of Legramanti et al. (Legramanti et al. 2020 Biometrika 107, 745-752. (doi:10.1093/biomet/asaa008)), which is a spike-and-slab shrinkage prior where the spike probability is stochastically increasing and constructed from the stick-breaking representation of a Dirichlet process prior. As a first contribution, this CUSP prior is extended by involving arbitrary stick-breaking representations arising from beta distributions. As a second contribution, we prove that exchangeable spike-and-slab priors, which are popular and widely used in sparse Bayesian factor analysis, can be represented as a finite generalized CUSP prior, which is easily obtained from the decreasing order statistics of the slab probabilities. Hence, exchangeable spike-and-slab shrinkage priors imply increasing shrinkage as the column index in the loading matrix increases, without imposing explicit order constraints on the slab probabilities. An application to sparse Bayesian factor analysis illustrates the usefulness of the findings of this paper. A new exchangeable spike-and-slab shrinkage prior based on the triple gamma prior of Cadonna et al. (Cadonna et al. 2020 Econometrics 8, 20. (doi:10.3390/econometrics8020020)) is introduced and shown to be helpful for estimating the unknown number of factors in a simulation study. This article is part of the theme issue 'Bayesian inference: challenges, perspectives, and prospects'.

Semiparametric finite mixture of regression models with Bayesian P-splines

Mixture models provide a useful tool to account for unobserved heterogeneity and are at the basis of many model-based clustering methods. To gain additional flexibility, some model parameters can be expressed as functions of concomitant covariates. In this Paper, a semiparametric finite mixture of regression models is defined, with concomitant information assumed to influence both the component weights and the conditional means. In particular, linear predictors are replaced with smooth functions of the covariate considered by resorting to cubic splines. An estimation procedure within the Bayesian paradigm is suggested, where smoothness of the covariate effects is controlled by suitable choices for the prior distributions of the spline coefficients. A data augmentation scheme based on difference random utility models is exploited to describe the mixture weights as functions of the covariate. The performance of the proposed methodology is investigated via simulation experiments and two real-world datasets, one about baseball salaries and the other concerning nitrogen oxide in engine exhaust.

Bayesian sparse convex clustering via global-local shrinkage priors

Sparse convex clustering is to group observations and conduct variable selection simultaneously in the framework of convex clustering. Although a weighted $$L_1$$ L 1 norm is usually employed for the regularization term in sparse convex clustering, its use increases the dependence on the data and reduces the estimation accuracy if the sample size is not sufficient. To tackle these problems, this paper proposes a Bayesian sparse convex clustering method based on the ideas of Bayesian lasso and global-local shrinkage priors. We introduce Gibbs sampling algorithms for our method using scale mixtures of normal distributions. The effectiveness of the proposed methods is shown in simulation studies and a real data analysis.