exact_horseshoe: Run exact MCMC algorithm for horseshoe prior

In Mhorseshoe: Approximate Algorithm for Horseshoe Prior

Run exact MCMC algorithm for horseshoe prior

Description

The exact MCMC algorithm for the horseshoe prior introduced in section 2.1 of Johndrow et al. (2020).

Usage

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.

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.

Value

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}.

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.

Examples

# 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

Mhorseshoe documentation built on April 12, 2025, 1:33 a.m.