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:

 

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