Samples variances from the horseshoe prior using Gruber & Kastner (2024) — sample_variances

Performs one Gibbs sampling iteration for the **exported hyper-parameters** #' of the horseshoe prior variance parameters: local variances ($\lambda_j^2$, argument theta), global variance ($\tau^2$, argument zeta), and their corresponding auxiliary variables ($\nu_j$ and $\varpi$, arguments nu and varpi). The horseshoe prior Carvalho, Polson, Scott (2010) is a continuous shrinkage prior for Bayesian variable selection with the hierarchical structure: $$\beta_j \sim N(0, \lambda_j^2 \tau^2)$$ $$\lambda_j \sim C^+(0, 1)$$ $$\tau \sim C^+(0, 1)$$ where $C^+(0, 1)$ denotes the half-Cauchy distribution, $\lambda_j^2$ are the local shrinkage parameters (argument theta), and $\tau^2$ is the global shrinkage parameter (argument zeta). The prior variance for coefficient $j$ is $V_{i,j} = \lambda_j^2 \tau^2$.

The half-Cauchy distributions are represented using auxiliary variables $\nu_j$ and $\varpi$ to facilitate Gibbs sampling. This implementation allows updating only a subset of coefficients specified by the indices in argument ind.

Usage

sample_variances_horseshoe(coefs, theta, zeta, nu, varpi)

Arguments

coefs: a $p$-vector with the current coefficient values $\beta_j$. C++: an arma::vec vector object.
theta: a $p$-vector with the local variance parameters $\lambda_j^2$, updated by reference. C++: an arma::vec vector object.
zeta: a numeric scalar with the global variance parameter $\tau^2$, updated by reference. C++: a double scalar.
nu: a $p$-vector with auxiliary variables for the local shrinkage, updated by reference. C++: an arma::vec vector object.
varpi: a numeric scalar with the auxiliary variable for the global shrinkage, updated by reference. C++: a double scalar.

Value

A **list** of C++ objects of class Rcpp::List containing the updated variance parameters. The elements are:

V_i: A $p$-vector of the total prior variances $\lambda_j^2 \tau^2$. R: a **numeric vector**. C++: an arma::vec object.
theta: The updated $p$-vector of local variances $\lambda_j^2$. R: a **numeric vector**. C++: an arma::vec object.
zeta: The updated scalar global variance $\tau^2$. R: a **positive scalar**. C++: a double scalar.
nu: The updated $p$-vector of local auxiliary variables $\nu_j$. R: a **numeric vector**. C++: an arma::vec object.
varpi: The updated scalar global auxiliary variable $\varpi$. R: a **positive scalar**. C++: a double scalar.

Details

This function is based on C++ code from the R package bayesianVARs by Gruber & Kastner (2024) and is using objects and commands from the armadillo library. Thanks to the RcppArmadillo package by Eddelbuettel, Francois, Bates, Ni, & Sanderson (2025).

References

Carvalho C.M., Polson N.G., Scott J.G. (2010). The horseshoe estimator for sparse signals. Biometrika, 97(2), 465-480. <doi:10.1093/biomet/asq017>

Eddelbuettel D., Francois R., Bates D., Ni B., Sanderson C. (2025). RcppArmadillo: 'Rcpp' Integration for the 'Armadillo' Templated Linear Algebra Library. R package version 15.0.2-2. <doi:10.32614/CRAN.package.RcppArmadillo>

Gruber L.,Kastner G. (2024). bayesianVARs: MCMC Estimation of Bayesian Vector Autoregressions. R package version 0.1.5, <doi:10.32614/CRAN.package.bayesianVARs>

Makalic E., Schmidt D.F. (2016). A Simple Sampler for the Horseshoe Estimator. IEEE Signal Processing Letters, 23(1), 179-182. <doi:10.1109/LSP.2015.2503725>

Sanderson C., Curtin R. (2025). Armadillo: An Efficient Framework for Numerical Linear Algebra. International Conference on Computer and Automation Engineering, 303-307, <doi:10.1109/ICCAE64891.2025.10980539>

Author

Longcan Li longcando@outlook.com

Examples

sample_variances_horseshoe( rep(0, 2), rep(0, 2), 1, rep(1, 2), 1)
#> $V_i
#>            [,1]
#> [1,]  0.8089962
#> [2,] 13.5056860
#> 
#> $theta
#>            [,1]
#> [1,]  0.7234863
#> [2,] 12.0781516
#> 
#> $zeta
#> [1] 1.118191
#> 
#> $nu
#>           [,1]
#> [1,] 46.755885
#> [2,]  1.299946
#> 
#> $varpi
#> [1] 16.30788
#>