HS.post.mean {horseshoe}

Posterior mean for the horseshoe for the normal means problem.

Description

Compute the posterior mean for the horseshoe for the normal means problem (i.e. linear regression with the design matrix equal to the identity matrix), for a fixed value of tau, without using MCMC, leading to a quick estimate of the underlying parameters (betas). Details on computation are given in Carvalho et al. (2010) and Van der Pas et al. (2014).

Usage

HS.post.mean(y, tau, Sigma2 = 1)

Arguments

y

The data. An n*1 vector.

tau

Value for tau. Warning: tau should be greater than 1/450.

Sigma2

The variance of the data.

Details

The normal means model is:

y_i=\beta_i+\epsilon_i, \epsilon_i \sim N(0,\sigma^2)

And the horseshoe prior:

\beta_j \sim N(0,\sigma^2 \lambda_j^2 \tau^2)

\lambda_j \sim Half-Cauchy(0,1).

If \tau and \sigma^2 are known, the posterior mean can be computed without using MCMC.

Value

The posterior mean (horseshoe estimator) for each of the datapoints.

References

Carvalho, C. M., Polson, N. G., and Scott, J. G. (2010), The horseshoe estimator for sparse signals. Biometrika 97(2), 465–480.

van der Pas, S. L., Kleijn, B. J. K., and van der Vaart, A. W. (2014), The horseshoe estimator: Posterior concentration around nearly black vectors. Electronic Journal of Statistics 8(2), 2585–2618.

See Also

HS.post.var to compute the posterior variance. See HS.normal.means for an implementation that does use MCMC, and returns credible intervals as well as the posterior mean (and other quantities). See horseshoe for linear regression.

Examples

#Plot the posterior mean for a range of deterministic values
y <- seq(-5, 5, 0.05)
plot(y, HS.post.mean(y, tau = 0.5, Sigma2 = 1))

#Example with 20 signals, rest is noise
#Posterior mean for the signals is plotted in blue
truth <- c(rep(0, 80), rep(8, 20))
data <-  truth + rnorm(100)
tau.example <- HS.MMLE(data, 1)
plot(data, HS.post.mean(data, tau.example, 1),
 col = c(rep("black", 80), rep("blue", 20)))