The Jeffreys' prior for data from an [[exponential distribution]] is $\pi(\lambda) \propto \frac{\sqrt{n}}{\lambda}$ ## derivation A likelihood of the exponential distribution (from the pdf) is $f(x) = \lambda e^{-\lambda x}$ Taking the log of the marginal pdf we get $f(x_i) = \log \lambda - \lambda x_i$ Taking the first derivative $\frac{\partial \log f(x_i)}{\partial \lambda} = \frac{1}{\lambda} - x_i$ Taking the second derivative $\frac{\partial^2 \log f(x_i)}{\partial^2 \lambda} = - \frac{1}{\lambda^2}$ The Fisher information is n times the negative expectation of the second derivative $I(\lambda) = -n E \Big [ \frac{\partial^2 \log f(x_i)}{\partial^2 \lambda} \Big ] = -n E \Big [ - \frac{1}{\lambda^2} \Big ]= \frac{n}{\lambda^2}$ Jeffreys' prior is the square root of the Fisher information $\pi(\lambda) \propto \sqrt{I(\lambda)} \propto \sqrt{\frac{n}{\lambda^2}} \propto \frac{\sqrt{n}}{\lambda}$