Relative Entropy Derivative Bounds

Abstract

We show that the derivative of the relative entropy with respect to its parameters is lower and upper bounded. We characterize the conditions under which this derivative can reach zero. We use these results to explain when the minimum relative entropy and the maximum log likelihood approaches can be valid. We show that these approaches naturally activate in the presence of large data sets and that they are inherent properties of any density estimation process involving large numbers of random variables.

Keywords

• relative entropy
• Kullback-Leibler divergence
• Shannon differential entropy
• asymptotic equipartition principle
• typical set
• Fisher information
• maximum log likelihood