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:

 

 

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