Abstract
A new information matrix [F] with elements F mn = 〈 (y m - a m )(y n - a n) (∂ ln p(y | a)/∂a m ) (∂ ln p(y | a)/∂a n ) 〉 is analyzed. The PDF p(y | a) is the usual likelihood law. [F] differs from the Fisher information matrix by the presence of the first two factors in the given expectation. These factors make F mn unitless, in contrast with the Fisher information. This lack of units allows F mn values from entirely different phenomena to be compared as, for example, Shannon information values can be compared. Each element F mn defines an error inequality analogous to the Cramer-Rao inequality. In the scalar case F mn ≡ F, for a normal p(y|a) law F = 3, while for an exponential law F = 9. A variational principle F = min (called FMIN) allows an unknown PDF p(x) to be estimated in the presence of weak information. Under certain conditions F obeys a “Boltzmann F-theorem” ∂F/∂t ⩽ 0, indicating that F is a physical entropy. Finally, the trace ℱ of [F] may be used as the scalar information quantity in an information-based principle for deriving distribution laws p of physics.
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Frieden, B.R. F-Information, a Unitless Variant of Fisher Information. Foundations of Physics 29, 1521–1541 (1999). https://doi.org/10.1023/A:1018806603833
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DOI: https://doi.org/10.1023/A:1018806603833