Suppose that conditional on $\tau$, the random variable $X$ has normal distribution with mean zero and variance $1/ \tau$. The prior distribution for $\tau$ is Gamma with parameter $\alpha$ and $\beta$. What is the posterior distribution of $\tau$?
I have been working on this and got stuck in how to get the pdf for $X$.
Thank you very much
The likelihood is $$ L(\tau) \propto \sqrt{\tau}\exp\left( \frac{-\tau x^2}{2} \right). $$ The prior is $$ f(\tau) \propto \tau^{\alpha-1} \exp\left(-\beta\tau\right) \, d\tau. $$ The posterior is therefore $$ L(\tau)f(\tau)\,d\tau \propto \tau^{\alpha-1/2}\exp\left( -\tau\left(\frac{x^2}{2}+\beta\right) \right) \,d\tau. $$
Thus the posterior is a Gamma distribution whose "shape parameter" is $\alpha+1/2$ and whose rate parameter $\beta+(x^2/2)$.
The question was ambiguous on one point: WHICH of the two frequently used parametrizations of the Gamma family was intended? Above I assumed $\beta$ was supposed to be the rate parameter. But if it was intended to be the scale parameter, then you'd have $-\tau/\beta$ instead of $-\beta\tau$, and the reciprocal of $(x^2/2) + (1/\beta)$ in place of $\beta$.