F-Information, a Unitless Variant of Fisher Information

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 of F] may be used as the scalar information quantity in an information-based principle for deriving distribution laws p of physics.