Optimal guessing under nonextensive framework and associated moment bounds

Abstract

We consider the problem of guessing the realization of a random variable but under more general Tsallis' non-extensive entropic framework rather than the classical Maxwell-Boltzman-Gibbs-Shannon framework. We consider both the conditional guessing problem in the presence of some related side information, and the unconditional one where no such side-information is available. For both types of the problem, the non-extensive moment bounds of the required number of guesses are derived; here we use the $q$-normalized expectation in place of the usual (linear) expectation to define the non-extensive moments. These moment bounds are seen to be a function of the logarithmic norm entropy measure, a recently developed two-parameter generalization of the Renyi entropy, and hence provide their information theoretic interpretation. We have also considered the case of uncertain source distribution and derived the non-extensive moment bounds for the corresponding mismatched guessing function. These mismatched bounds are interestingly seen to be linked with an important robust statistical divergence family known as the relative $(\alpha,\beta)$-entropies; similar link is discussed between the optimum mismatched guessing with the extremes of these relative entropy measures.