Jeffreys prior

In Bayesian probability, the Jeffreys prior, named after Sir Harold Jeffreys, is a non-informative (objective) prior distribution for a parameter space; it is proportional to the square root of the determinant of the Fisher information:

It has the key feature that it is invariant under reparameterization of the parameter vector ${\displaystyle {\vec {\theta }}}$. This makes it of special interest for use with scale parameters.

For an alternative parameterization ${\displaystyle \varphi }$ we can derive

from

using the change of variables theorem for transformations and the definition of Fisher information:

For an alternative parameterization ${\displaystyle {\vec {\varphi }}}$ we can derive

...
Wikipedia