For data from [[exponential distribution]], with its rate parameter $\lambda$, let the [[prior]] distribution on $\lambda$ be a [[gamma distribution]], then the [[posterior]] distribution will also have the gamma distribution. $\Gamma(\alpha, \beta) \to \Gamma(\alpha + n, \beta + n \bar x)$ **Likelihood** $ L(\lambda | \mathbf{x}) = \lambda^n e^{-\lambda \sum x_i} $ **Prior** $ \pi(\lambda) \propto \lambda^{\alpha-1} e^{-\beta \lambda} $ **Posterior** $ \pi(\lambda | \mathbf{x}) \propto \lambda^{(\alpha+n)-1} e^{-(\beta + \sum x_i) \lambda} $ See the [[Jeffreys' prior for the exponential distribution]].