Gamma Distribution

$gamma(r,v)$

$y>0,r>0,v>0$

Inverse Gamma Prior

If $y\sim gamma(r,v)$ find the pdf of $W=\frac{1}{y}$

Using Change of Variables
$⟹y=\frac{1}{W}⟹\frac{dy}{dW}=\frac{1}{W^2}$

Using CDF technique:

$P(W\le w)=P(\frac{1}{Y}\le w)=P(\frac{1}{w}\le Y)=1-P(Y<\frac{1}{w})=1-F_y\left(\frac{1}{w}\right)$

Inverse Gamma to Chi-Squared

Case: $W>0$

Variance Posterior

Likelihood $P(y_i|{\sigma }^2)={\left[\frac{1}{2\pi {\sigma }^2}\right]}^{1/2}\exp \left\{\frac{-(y_i-\mu {)}^2}{2{\sigma }^2}\right\}={\left[\frac{1}{2\pi }\right]}^{1/2}({\sigma }^2{)}^{-1/2}\exp \left\{\frac{-(y_i-\mu {)}^2}{2{\sigma }^2}\right\}$
For multiple observations:
$P(\{y_i\}|{\sigma }^2)={\left[\frac{1}{2\pi }\right]}^{n/2}({\sigma }^2{)}^{-n/2}\exp \left\{\frac{\sum _{}^{}-(y_i-\mu {)}^2}{2}\left(\frac{1}{{\sigma }^2}\right)\right\}$

Inverse Gamma $r_{\ast }=\frac{n}{2}+r$ and $v_{\ast }=\frac{\sum _{}^{}(y_i-\mu {)}^2}{2}+v$

Example

The strength of an item is known to be Normally distributed with mean 200 and unknown variance ${\sigma }^2$. A random sample of ten items is taken and their strength measured. The strengths are:
215 186 216 203 221 188 202 192 208 195

Uniform prior $p({\sigma }^2)\propto 1$

inv. gamma($r_{\ast }=\frac{\stackrel{k_{\ast }}{\overbrace{8}}}{2},v_{\ast }=\frac{\stackrel{S_{\ast }}{\overbrace{1428}}}{2}$)

by chi theorem $W=\frac{S_{\ast }}{{\sigma }^2}\sim {\chi }^2(df=k\ast )$

where we are doing area to the left for 0.025 and and 0.975.

For $H_0:\sigma \le 8⟹{\sigma }^2\le 64$

reject $H_0$

Example

Using Jefferys' Prior $p({\sigma }^2)\propto \frac{1}{{\sigma }^2}=({\sigma }^2{)}^{-1}$