# HS.post.mean: Posterior mean for the horseshoe for the normal means... In horseshoe: Implementation of the Horseshoe Prior

## 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

 `1` ```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=β_i+ε_i, ε_i \sim N(0,σ^2)

And the horseshoe prior:

β_j \sim N(0,σ^2 λ_j^2 τ^2)

λ_j \sim Half-Cauchy(0,1).

If τ and σ^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.

`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.
 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```#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))) ```