Jeffreys prior

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

<math>pleft(vecthetaright) propto sqrt.,</math>

It has the key feature that it is invariant under reparameterization of the parameter vector <math>vectheta</math>. This makes it of special interest for use with scale parametersJaynes, E. T. (1968) "Prior Probabilities", IEEE Trans. on Systems Science and Cybernetics, SSC-4, 227 ..


For an alternate parameterization <math>vecvarphi</math> we can derive

<math>p(vecvarphi) propto sqrt,</math>


<math>p(vectheta) propto sqrt,</math>

using the change of variables theorem, the definition of Fisher information, and that the product of determinants is the determinant of the matrix product:

beginp(vecvarphi) & = p(vectheta) left|detfracright| \& propto sqrt \& = sqrt \& = sqrt \& = sqrt= sqrt.end</math>

In the simpler case of a single parameter space variable we can derive

beginp(varphi) & = p(theta) left|fracright|propto sqrt \& = sqrt= sqrt= sqrt.end</math>


From a practical and mathematical standpoint, a valid reason to use this non-informative prior...
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