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I'm trying to teach myself Bayesian statistics and am currently trying find the posterior distribution on the following problem:

Suppose $X_1,...,X_n$ are iid exponential random variables, with density $f(x;\theta) =\theta e^{-\theta x}$ ,and let us suppose that we have a prior on $\theta$ with density $\theta^{\alpha -1}e^{-\beta \theta}$

Now, to find the posterior, we do Likelihood times Prior. The likelihood is given by

$$\theta^n e^{-n^2\theta \bar{x}}$$

So that we have

$$\theta^{\alpha -1}e^{-\beta \theta}\theta^n e^{-n^2\theta \bar{x}}=\theta^{n+\alpha-1}e^{-n^2 \theta \bar{x} - \beta \theta}$$

And in the textbook I am reading, from here you are meant to deduce the posterior distribution, but I cannot see an obvious distribution. Have I made an error? Can the posterior density function contain the random variable (x in this case)?

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    $\begingroup$ The posterior on $\theta$ is the conditional on $x$, so it is natural to find $x$ in the density. $\endgroup$
    – Xi'an
    Oct 18, 2020 at 5:57
  • $\begingroup$ @Xi'an do we still treat $x$ as a random variable in this case? Has it been observed as the posterior is conditional on $x$? $\endgroup$
    – Bill
    Oct 18, 2020 at 9:06

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Your likelihood should have $n$ instead of $n^2$. Anyway, the function you found is proportional to the pdf of a Gamma Distribution.

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