Jeffreys prior

Remarks

Example: for one Bernoulli observation with chance $p \in (0, 1)$, $I(p) = 1/(p(1 - p))$. So the Jeffreys prior is proportional to $p^{-1/2}(1 - p)^{-1/2}$, which is the proper distribution $\mathrm{Beta}(1/2, 1/2)$. For the mean $\mu$ of $\mathcal{N}(\mu, \sigma^2)$ with $\sigma$ known, $I(\mu) = 1/\sigma^2$ is constant, so the Jeffreys prior is the improper flat prior on $\R$. For the scale $\sigma > 0$ of $\mathcal{N}(0, \sigma^2)$, $I(\sigma) = 2/\sigma^2$, so the Jeffreys prior is proportional to $1/\sigma$. Invariance: let $\varphi = h(\theta)$ for a smooth bijection $h$ with smooth inverse. Then the Jeffreys prior computed in the parameter $\varphi$ is the image of the Jeffreys prior in $\theta$. The reason is that $\det I$ is multiplied by the square of the Jacobian determinant of the change of variables. Non-example: the flat prior on $p \in (0, 1)$ is not the Jeffreys prior of the Bernoulli model, since its density $1$ is not proportional to $p^{-1/2}(1 - p)^{-1/2}$.