Variance of Entropy Decreases Under the Polar Transform

We consider the behavior of entropy of data elements as they are operated on by polar transforms. The data elements under consideration are pairs (X,Y ) where X is a binary random variable and Y is some side information about X . The entropy random variable for such a data element is defined as h(X|Y ) = − log pX|Y (X|Y ). The variance of entropy (varentropy) is defined as Var(h(X|Y )). A polar transform of order two is a mapping that takes two independent data elements and produces two new data elements (that are in general correlated). We show that the sum of the varentropies of the output data elements is less than or equal to the sum of the varentropies of the input data elements, with equality if and only if at least one of the input data elements has zero varentropy. This result is then extended to polar transforms of higher orders and an asymptotic analysis is given for an important special case in which the average output varentropy decreases monotonically to zero as the transform size increases.