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?
Best Answer
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., $\theta\mid x\sim\operatorname{Gamma}(\alpha=\frac12+\sum x_i,\beta=n)$.