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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:




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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