# Posterior distribution of normal with gamma prior on the precision

Find the posterior distribution when

$$x|\sigma\sim \mathcal N(0,\sigma^2),\:\:\: 1/\sigma^2\sim \mathsf{Gamma}(1,2)$$

I'm stuck in this exercise, I know that $$\pi(x|\sigma)\approx f(x|\sigma)\pi(\sigma)\cdot\frac{1}{m(x)}$$

Maybe I am thinking wrong, but I would not have to find a prior of $\sigma$?

• Is that the quotient of $x, \sigma$, or $x$ given $\sigma$? – Sean Easter Mar 10 '16 at 19:19
• @SeanEaster the pdf is given and the other is $\frac{1}{\sigma^2}$ – user72621 Mar 10 '16 at 19:20
• Please add the [self-study] tag & read its wiki. – gung - Reinstate Monica Mar 10 '16 at 21:49
• @amoeba, I believe I voted to close as SS, although I don't really remember anymore. You can vote to reopen if you like. – gung - Reinstate Monica Mar 17 '16 at 20:31
• Could you explain what the symbols "$m$", "$f$", and "$\pi$" mean to you? – whuber Mar 28 '16 at 14:51

In effect, you know the prior on $\sigma^2$: It's inverse gamma. After expressing the posterior as the product of normal likelihood and inverse-gamma prior, one can manipulate the posterior until it's recognizable as another inverse gamma. (Left as an exercise, but confirmed here.)