horseshoe (version 0.2.0)

HS.post.var: Posterior variance for the horseshoe for the normal means problem.

Description

Compute the posterior variance 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. Details on computation are given in Carvalho et al. (2010) and Van der Pas et al. (2014).

Usage

HS.post.var(y, tau, Sigma2)

Arguments

y

The data. An \(n*1\) vector.

tau

Value for tau. Tau should be greater than 1/450.

Sigma2

The variance of the data.

Value

The posterior variance for each of the datapoints.

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 variance can be computed without using MCMC.

References

Carvalho, C. M., Polson, N. G., and Scott, J. G. (2010), The horseshoe estimator for sparse signals. Biometrika 97(2), 465<U+2013>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<U+2013>2618.

See Also

HS.post.mean to compute the posterior mean. 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

# NOT RUN {
#Plot the posterior variance for a range of deterministic values
y <- seq(-8, 8, 0.05)
plot(y, HS.post.var(y, tau = 0.05, Sigma2 = 1))

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

# }