Probability and Relative Derivative d/dy ln(P)

For the case of physical quantities with units, such as mass, time, length etc, a change in the value of the quantity makes sense by itself in addition to a relative change. In other words, one does not need to know the mass of an object to understand what a change of 1 kg is. Probability, on the other hand, is a strictly relative quantity and so needs to be normalized. If it is not normalized, one needs to know relative probabilities. To say that an unnormalized probability changes by a value of 10 does not mean anything. Given that equations often solve for an unnormalized probability function, we argue that the notion of the relative probability should appear in equations. Even in the case of ln(P(y)), called information in information theory, for which there is no derivative, taking a derivative, say d/dy, immediately creates a relative probability derivative. Thus the form ln(P) is in keeping with the notion of a relative derivative.

    Here we examine the idea of a relative derivative and its association with information ln(P), Fisher information, quantum mechanics and the Fokker-Planck equation as well as classical statistical balance in the presence of a potential.