What is the posterior distribution for parameter $b$ with $X \sim Gamma(a,b)$, under the Jeffreys prior? We can assume that $a$ is known.
The Jeffreys prior is the square of the Fisher information of $b$:
$p(b)=\frac{\sqrt(a)}{b}$.
Then using Bayes' rule we have
$p(b|x) \propto p(x|b) \,p(b) = \dfrac{b^a}{\Gamma (a)}x^{a-1}e^{-xb}\cdot\frac{\sqrt(a)}{b}$
Next we look for the kernel of a Gamma distribution. But this is where I am stuck.
What is the next step for deriving the posterior distribution for $b$?
Best Answer
It is difficult to perceive where you get stuck: $$\begin{align*}p(b|x)&\propto \dfrac{b^a}{\Gamma (a)}x^{a-1}e^{-xb}\cdot\dfrac{\sqrt{a}}{b} \\ &\propto b^{a-1} e^{-xb} \\ &\propto \dfrac{x^a\,b^{a-1}}{\Gamma(a)}\,e^{-xb}\end{align*}$$ which shows the posterior is a Gamma $\mathcal{G}(a,x)$ distribution.