dist_horseshoe: The Horseshoe distribution

Description

The horseshoe distribution (Carvalho et al., 2008) is a heavy-tailed continuous distribution defined as a scale mixture of normals. It is primarily used as a shrinkage prior in sparse Bayesian regression, where it concentrates mass near zero while retaining heavy tails that leave large signals unshrunk.

Usage

dist_horseshoe(lambda, tau)

Arguments

lambda: A positive numeric vector of local scale parameters $\lambda > 0$ (one per observation).
tau: A positive scalar global scale parameter $\tau > 0$.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_horseshoe.html

In the following, let $X$ be a horseshoe random variable with local scale parameter lambda = $\lambda > 0$ and global scale parameter tau = $\tau > 0$.

Support: $x \in \mathbb{R}$, the set of all real numbers.

Mean: $E(X)$ — not available in closed form.

Variance: $\mathrm{Var}(X)$ — not available in closed form.

Probability density function (p.d.f):

The horseshoe density does not have a simple closed form but can be expressed as a scale mixture:

$$ X \mid \lambda, \tau \sim \mathcal{N}(0,\, \lambda^2 \tau^2) $$

where the half-Cauchy hyperprior $\lambda \sim C^+(0, 1)$ induces the characteristic horseshoe shrinkage behaviour.

References

Carvalho, C.M., Polson, N.G., and Scott, J.G. (2008). "The Horseshoe Estimator for Sparse Signals". Discussion Paper 2008-31. Duke University Department of Statistical Science.

Carvalho, C.M., Polson, N.G., and Scott, J.G. (2009). "Handling Sparsity via the Horseshoe". Journal of Machine Learning Research, 5, p. 73–80.

Examples

Run this code

dist <- dist_horseshoe(lambda = c(0.5, 1, 2), tau = 1)
dist

if (FALSE) { # requireNamespace("LaplacesDemon", quietly = TRUE)
support(dist)
generate(dist, 10)

density(dist, 0)
density(dist, 0, log = TRUE)
}