exact_horseshoe {Mhorseshoe}

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,
  a = 1/5,
  b = 10,
  s = 0.8,
  tau = 1,
  sigma2 = 1,
  w = 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.

a

A tuning parameter of the rejection sampler, where the default value is a = 1/5.

b

A tuning parameter of the rejection sampler, where the default value is b = 10.

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.

tau

Initial value of the global shrinkage parameter \tau when starting the algorithm. The default is 1.

sigma2

Initial value of error variance \sigma^{2}. The default is 1.

w

A hyperparameter of gamma prior for \sigma^{2}. The default is 1.

alpha

100(1-\alpha)\% credible interval setting argument.

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