# Run exact MCMC algorithm for horseshoe prior The exact MCMC algorithm for the horseshoe prior introduced in section 2.1 of Johndrow et al. (2020). ```r exact_horseshoe( y, X, burn = 1000, iter = 5000, tau = 1, s = 0.8, sigma2 = 1, alpha = 0.05, ... ) ``` ## Arguments - `y`: Response vector, $y \in \mathbb{R}^{N}$. - `X`: Design matrix, $X \in \mathbb{R}^{N \times p}$. - `burn`: Number of burn-in samples. The default is 1000. - `iter`: Number of samples to be drawn from the posterior. The default is 5000. - `tau`: Initial value of the global shrinkage parameter $\tau$ when starting the algorithm. The default is 1. - `s`: $s^{2}$ is the variance of tau's MH proposal distribution. 0.8 is a good default. If set to 0, the algorithm proceeds by fixing the global shrinkage parameter $\tau$ to the initial setting value. - `sigma2`: Initial value of error variance $\sigma^{2}$. The default is 1. - `alpha`: $100(1-\alpha)\%$ credible interval setting argument. - `...`: There are additional arguments **threshold**, **a**, **b**, and **w**. **a** and **b** are arguments of the internal rejection sampler function, and the default values are $a = 1/5,\ b = 10$. **w** is the argument of the prior for $\sigma^{2}$, and the default value is $w = 1$. ## Returns - **BetaHat**: Posterior mean of $\beta$. - **LeftCI**: Lower bound of $100(1-\alpha)\%$ credible interval for $\beta$. - **RightCI**: Upper bound of $100(1-\alpha)\%$ credible interval for $\beta$. - **Sigma2Hat**: Posterior mean of $\sigma^{2}$. - **TauHat**: Posterior mean of $\tau$. - **LambdaHat**: Posterior mean of $\lambda_{j},\ j=1,2,...p.$. - **BetaSamples**: Samples from the posterior of $\beta$. - **LambdaSamples**: Lambda samples through rejection sampling. - **TauSamples**: Tau samples through MH algorithm. - **Sigma2Samples**: Samples from the posterior of the parameter $sigma^{2}$. ## Details The exact MCMC algorithm introduced in Section 2.1 of Johndrow et al. (2020) is implemented in this function. This algorithm uses a blocked-Gibbs sampler for $(\tau, \beta, \sigma^2)$, where the global shrinkage parameter $\tau$ is updated by an Metropolis-Hastings algorithm. The local shrinkage parameter $\lambda_{j},\ j = 1,2,...,p$ is updated by the rejection sampler. ## Examples ```r # Making simulation data. set.seed(123) N <- 50 p <- 100 true_beta <- c(rep(1, 10), rep(0, 90)) X <- matrix(1, nrow = N, ncol = p) # Design matrix X. for (i in 1:p) { X[, i] <- stats::rnorm(N, mean = 0, sd = 1) } y <- vector(mode = "numeric", length = N) # Response variable y. e <- rnorm(N, mean = 0, sd = 2) # error term e. for (i in 1:10) { y <- y + true_beta[i] * X[, i] } y <- y + e # Run exact_horseshoe result <- exact_horseshoe(y, X, burn = 0, iter = 100) # posterior mean betahat <- result$BetaHat # Lower bound of the 95% credible interval leftCI <- result$LeftCI # Upper bound of the 95% credible interval RightCI <- result$RightCI ``` ## References Johndrow, J., Orenstein, P., & Bhattacharya, A. (2020). Scalable Approximate MCMC Algorithms for the Horseshoe Prior. In Journal of Machine Learning Research, 21, 1-61.

exact_horseshoe function