# Jeffreys Prior and Posterior

The question says provided a random sample $$(x_1, x_2, \ldots ,x_n)=x$$ from a Poisson distribution say $$P(\theta)$$.

It asks to find the Jeffreys prior distribution for $$\theta$$ and then find the posterior distribution of $$\theta|x$$.

I found the Jeffreys prior but have a doubt on the 2nd part of the question. What I know is given a Poisson prior, we would find the posterior distribution which will usually be a Gamma distribution. But they didn't provide other distributions in the question. How can we find the posterior distribution?

The Jeffreys' (improper) prior for a $$\operatorname{Poisson}(\theta)$$ is $$p_{\mathrm{prior}}(\theta)\propto \theta^{-1/2} 1_{\theta>0}.$$ You are given $$x_i\sim\operatorname{Poisson}(\theta)$$, so assuming $$x_i$$s are independent(!), the posterior is $$p_{\mathrm{posterior}}(\theta\mid x)\propto p(x\mid\theta)p_{\mathrm{prior}}(\theta) \propto e^{-n\theta}\theta^{-1/2+\sum x_i}1_{\theta>0}$$ i.e., $$\theta\mid x\sim\operatorname{Gamma}(\alpha=\frac12+\sum x_i,\beta=n)$$.