91Ó°ÊÓ

ξ(θ)be a p.d.f. that is defined as follows for constants

α >0 andβ >0:

\(\xi \left( \theta \right) = \left\{ \begin{aligned}{l}\frac{{{\beta ^\alpha }}}{{\Gamma \left( \alpha \right)}}{\theta ^{ - \left( {\alpha + 1} \right)}}{e^{ - \beta /\theta }}\,\,\,for\,\,\theta > 0\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,for\,\,\theta \le 0\end{aligned} \right.\)

A distribution with this p.d.f. is called an inverse gamma

distribution.

If X has the normal distribution with a known value of the mean\(\mu \)and an unknown value of the standard deviation\(\sigma \), then the p.d.f of X has the form

\(f\left( {x|\sigma } \right) \propto \frac{1}{\sigma }\exp \left( { - \frac{{{{\left( {x - \mu } \right)}^2}}}{{2{\sigma ^2}}}} \right)\)

Therefore, if the prior p.d.f\(\xi \left( \theta \right)\)has the form

\(\xi \left( \theta \right) \propto {\sigma ^{ - a}}\exp \left( { - b/{\sigma ^2}} \right),\)

Then the posterior p.d.f of\(\sigma \)will also have the same form, with a replaced by a+1 and b replaced by\(b + {\left( {x - \mu } \right)^2}/2\)

Therefore,

\(\int_0^\infty {{\sigma ^{ - a}}\exp \left( { - b/{\sigma ^2}} \right)} d\sigma = \frac{1}{2}\int_0^\infty {{y^{\left( {a - 3} \right)/2}}\exp \left( { - by} \right)dy} \)

The integral will be finite if a>1 and b>0, and its value will be

\(\frac{{\Gamma \left( {\frac{1}{2}\left( {a - 1} \right)} \right)}}{{{b^{\left( {a - 1} \right)/2}}}}\)

Hence, for a>1 and b>0, the following function will be a p.d.f for\(\sigma > 0\)

\(\xi \left( \sigma \right) = \frac{{2{b^{\left( {a - 1} \right)/2}}}}{{\Gamma \left( {\frac{1}{2}\left( {a - 1} \right)} \right)}}{\sigma ^{ - a}}\exp \left( { - b/{\sigma ^2}} \right)\)

Finally, a more standard form for this p.d.f, by replacing a and b by\(\alpha = \left( {a - 1} \right)/2\,\,{\rm{and}}\,\,\beta = b\)

\(\xi \left( \sigma \right) = \frac{{2{\beta ^\alpha }}}{{\Gamma \left( \alpha \right)}}{\sigma ^{ - \left( {2\alpha + 1} \right)}}\exp \left( { - \beta /{\sigma ^2}} \right)\,\,\,\,for\,\,\sigma > 0\)

The family of distributions for which the p.d.f has this form, for all values of \(\alpha > 0\,\,and\,\,\beta > 0\) , will be a conjugate family of the prior distribution.