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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:

 

l(X|\sigma ^{2})\propto (\sigma ^{2})^{-\frac{n}{2}}exp\Big(-\frac{nV}{2\sigma ^{2}}\Big)


 

where V is the sample variance V=\frac{1}{n}\displaystyle\sum_{i-1}^{n}(x_{i}-\mu)^{2}

 

With the uninformed prior:

 

\pi(\sigma^{2})\propto \frac{1}{\sigma^{2}}


 

this gives a posterior distribution of:

 

       

f(\sigma ^{2}|X)\propto (\sigma ^{2})^{-\big(\frac{n}{2}+1\big)}exp\Big(-\frac{nV}{2\sigma^{2}}\Big)~~~~\text{(1)}

   

If a variable X = Gamma(0,b,a), then the variable Y=1/X has the Inverse-Gamma density:

 

 

f(y) \propto y^{-(\propto +1)}exp\Big(-\frac{1}{\beta y}\Big)~~~~\text{(2)}

                                     

Comparing Equations 1 and 2 we see that:

 


\sigma^{2}=\Big[Gamma\Big(0,\frac{2}{nV},\frac{n}{2}\Big)\Big]^{-1}=nv\Big[Gamma\Big(0,2,\frac{n}{2}\Big)\Big]^{-1}=\frac{nV}{\chi^{2}(n)}


 

The last identity comes from here. Rearranging gives:

 


\sigma =\sqrt{\frac{\displaystyle\sum_{i-1}^{n}(x_{i}-\mu)^{2}}{\chi^{2}(n)}}


 


 

 

 


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