For data from the [[Poisson distribution]] $X \overset{iid} \sim Pois(\lambda)$, let the [[prior]] distribution on $\lambda$ be a [[gamma distribution]], then the [[posterior]] distribution will also have the gamma distribution. $\lambda \sim \Gamma(a, b) \to \lambda \sim \Gamma(a + n \bar x, \beta + n)$ **Prior** $f(\lambda) \propto \lambda^{\alpha-1} e^{-\beta \lambda}$ **Likelihood** $f(\lambda|x) \propto \lambda ^{n \bar x} e^{-n\lambda}$ **Posterior** $f(\lambda | x) \propto \lambda^{n \bar x + \alpha-1} e^{-\lambda(\beta + n)}$ See the [[Jeffreys' prior for the Poisson distribution]].