How to compute the variance for a bayesian estimator I can't figure out how to compute the variance of an estimator which is the mean of the posterior distribution let's say Gamma($\sum x_i+3, n+a$)
How to find out the variance of this mean ?
 A: In the Bayesian paradigm, distributions of interest are uncertainty distributions of unknown parameters.  So if you have a posterior distribution $f(\theta)$ for parameter $\theta$ you can get an uncertainty (credible) interval for $\theta$.  This interval is "summary measure agnostic" since it does not refer to the use of a point estimate summary for $f$ such as the posterior median, mean, or mode.  The posterior mean is a convenient posterior distribution point estimate but doesn't play a central role unless you are doing a formal loss function-based analysis and your loss function is the squared error.
The bottom line: concern yourself with uncertainty about the primary parameter of interest: $\theta$, not with some convenient point summary of an entire posterior distribution.
A: *

*variance estimation via samples - converges to the real variance with a lot of samples.

*closed formula for variance is little bit harder. It is possible only if you use a conjugate prior. In this wiki you will find formulas for the posterior distribution, given some prior - for example gamma dist is the conjugate prior for Poisson dist. the gamma dist itself has different conjugate priors, listed in the wiki. After having a closed-form formula for the posterior distribution, you can find its mean/variance/mode/etc.

