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.
Highlights
The divergence or relative entropy between two probability densities is a measure of dissimilarity between them
The most well known divergence approach is due to Kullback and Leibler which will be discussed in more detail below
It would be possible to replace one model with another since there would be a similarity between them for large parameter sets. This idea will be pursued in this paper by making use of fractional calculus to obtain a fractional form for the Kullback–Leibler divergence
Summary
The divergence or relative entropy between two probability densities is a measure of dissimilarity between them. The Kullback–Leibler divergence is not symmetric for large separations between densities and does not obey the triangle inequality. It is worth noting that there is a mathematical duality between the Kullback–Leibler divergence and the geodesic approach of information geometry The latter is more complicated to work with in the mathematical sense because, in many cases, the geodesic must be obtained via the solution of partial differential equations. Replacing one model (density) by another only for certain unique or restricted values in their parameters is not very useful for modelling physical processes or systems This is the inherent problem associated with the current form of any divergence method. It would be possible to replace one model with another since there would be a similarity between them for large parameter sets This idea will be pursued in this paper by making use of fractional calculus to obtain a fractional form for the Kullback–Leibler divergence
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