Bayesian estimate of the standard deviation of a Normal distribution with known mean | Vose Software

# Bayesian estimate of the standard deviation of a Normal distribution with known mean

The likelihood function for n observations from a Normal distribution is given by the product of the Normal probability densities for each sample:

where V is the sample variance

With the uninformed prior:

this gives a posterior distribution of:

(1)

If a variable X = Gamma(a,b), then the variable Y=1/X has the Inverse-Gamma density:

(2)

Comparing Equations 1 and 2 we see that:

The last identity comes from here. Rearranging gives: