Fractional Divergence of Probability Densities

Abstract

Abstract:The divergence or relative entropy between probability densities is examined. Solutions that minimise the divergence between two distributions are usually "trivial" or unique. By using a fractional-order formulation for the divergence with respect to the parameters, the distance between probability densities can be minimised so that multiple non-trivial solutions can be obtained. As a result, the fractional divergence approach reduces the divergence to zero even when this is not possible via the conventional method. This allows replacement of a more complicated probability density with one that has a simpler mathematical form for more general cases.

One-Parameter Weibull-Type Distribution, Its Relative Entropy with Respect to Weibull and a Fractional Two-Parameter Exponential Distribution

“…Observe that the m = 0 order gives < x 0 >= 1 which is also readily seen from (18), that is, the second axiom of probability is obtained which states that the integral of the density over the entire domain is unity. The expectation or mean is given by the m = 1 order and from (20)…”

“…Fractional mathematics has the potential to change the way statistical analysis of problems is made. The reader is referred to [16][17][18] for details and the references therein. It is worth noting that the word 'fractional' is a historical misnomer.…”

The fractional Kullback–Leibler divergence

“…The word 'fractional' should be understood as meaning generalised. Fractional mathematics has been used in the study of many problems [24][25][26][27][28][29][30][31][32][33][34]. Notation-wise, the fractional order derivative operator is sometimes written as d α /dx α and the fractional integral operator as d −α /dx −α .…”

“…The fractional divergence between densities has been studied before but this was done by obtaining fractional forms for the parameters ξ m of the densities [27]. Here the approach will be to derive a fractional version of the K-L divergence.…”

“…Either there is no detection (null hypothesis H 0 ) or there is (alternative hypothesis H 1 ). The K-L divergence can be used as an outcome detector [27]. Suppose that two exponential densities have parameters that are known or exact: u 0 = 1 and v 0 = 1/2:…”