Sparsity information and regularization in the horseshoe and other shrinkage priors

The horseshoe prior has proven to be a noteworthy alternative for sparse Bayesian estimation, but has previously suffered from two problems. First, there has been no systematic way of specifying a prior for the global shrinkage hyperparameter based on the prior information about the degree of sparsity in the parameter vector. Second, the horseshoe prior has the undesired property that there is no possibility of specifying separately information about sparsity and the amount of regularization for the largest coefficients, which can be problematic with weakly identified parameters, such as the logistic regression coefficients in the case of data separation. This paper proposes solutions to both of these problems. We introduce a concept of effective number of nonzero parameters, show an intuitive way of formulating the prior for the global hyperparameter based on the sparsity assumptions, and argue that the previous default choices are dubious based on their tendency to favor solutions with more unshrunk parameters than we typically expect a priori. Moreover, we introduce a generalization to the horseshoe prior, called the regularized horseshoe, that allows us to specify a minimum level of regularization to the largest values. We show that the new prior can be considered as the continuous counterpart of the spike-and-slab prior with a finite slab width, whereas the original horseshoe resembles the spike-and-slab with an infinitely wide slab. Numerical experiments on synthetic and real world data illustrate the benefit of both of these theoretical advances.