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

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

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

With the uninformed prior:

this gives a posterior distribution of:

Expanding the exponent:

where  is the sample variance

To get to the marginal posterior distribution for s2 we have to average this joint distribution over all m:

The only component in m is:

which, with a normalizing factor of   would be a Normal distribution density.  Thus the integral is just the reciprocal of this factor and we get:

(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. Finally: