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

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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: