Posterior distribution of normal with gamma prior on the precision

Question

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 $\sigma$?

@SeanEaster the pdf is given and the other is $\frac{1}{\sigma^2}$ — , Mar 10 '16 at 19:20
Please add the `[self-study]` tag & read its [wiki](http://stats.stackexchange.com/tags/self-study/info). — 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

score 1 · Accepted Answer · edited Mar 02 '22 at 13:13

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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 in Michael I. Jordan - The Conjugate Prior for the Normal Distribution)

edited Mar 02 '22 at 13:13

Royi

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answered Mar 10 '16 at 20:07

Sean Easter

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Posterior distribution of normal with gamma prior on the precision

1 Answers1