postmean {EbayesThresh}

Posterior mean estimator

Description

Given a single value or a vector of data and sampling standard deviations (sd equals 1 for Cauchy prior), find the corresponding posterior mean estimate(s) of the underlying signal value(s).

Usage

postmean(x, s, w = 0.5, prior = "laplace", a = 0.5)
postmean.laplace(x, s = 1, w = 0.5, a = 0.5)
postmean.cauchy(x, w)

Arguments

x

A data value or a vector of data.

s

A single value or a vector of standard deviations if the Laplace prior is used. If a vector, must have the same length as x. Ignored if Cauchy prior is used.

w

The value of the prior probability that the signal is nonzero.

prior

Family of the nonzero part of the prior; can be "cauchy" or "laplace".

a

The scale parameter of the nonzero part of the prior if the Laplace prior is used.

Value

If x is a scalar, the posterior mean E(\theta|x) where \theta is the mean of the distribution from which x is drawn. If x is a vector with elements x_1, ... , x_n and s is a vector with elements s_1, ... , s_n (s_i is 1 for Cauchy prior), then the vector returned has elements E(\theta_i|x_i, s_i), where each x_i has mean \theta_i and standard deviation s_i, all with the given prior.

Note

If the quasicauchy prior is used, the argument a and s are ignored.

If prior="laplace", the routine calls postmean.laplace, which finds the posterior mean explicitly, as the product of the posterior probability that the parameter is nonzero and the posterior mean conditional on not being zero.

If prior="cauchy", the routine calls postmean.cauchy; in that case the posterior mean is found by expressing the quasi-Cauchy prior as a mixture: The mean conditional on the mixing parameter is found and is then averaged over the posterior distribution of the mixing parameter, including the atom of probability at zero variance.

Author(s)

Bernard Silverman

References

See ebayesthresh and http://www.bernardsilverman.com

See Also

postmed

Examples

postmean(c(-2,1,0,-4,8,50), w = 0.05, prior = "cauchy")
postmean(c(-2,1,0,-4,8,50), s = 1:6, w = 0.2, prior = "laplace", a = 0.3)