Horsehoe prior

We can define the Horseshoe prior in the general context of a gaussian scale mixture model :

\begin{align*} z_i &\sim \operatorname{Normal}(x_i^T\beta,\; \sigma^2\, \omega_i^2)\\ \sigma^2 &\sim \pi(\sigma^2)\, \mathrm{d}\sigma^2\\ \omega_i &\sim \pi(\omega_i)\, \mathrm{d}\omega_i\\ \beta_j &\sim \operatorname{Normal}(0,\; \lambda_j^2\, \tau^2\, \sigma^2)\\ \lambda_{j} &\sim \operatorname{C}^{+}(0, 1)\\ \tau &\sim \operatorname{C}^+(0,1)\\ \end{align*}

It is characterized by the global (\(\tau\)) and local (\(\lambda_j\)) shrinkage parameters that follow a half-Cauchy distribution.