Deformed statistics formulation of the information bottleneck method

The theoretical basis for a candidate variational principle for the information bottleneck (IB) method is formulated within the ambit of the generalized nonadditive statistics of Tsallis. Given a nonadditivity parameter q, the role of the additive duality of nonadditive statistics (q* = 2 - q ) in relating Tsallis entropies for ranges of the nonadditivity parameter q < 1 and q > 1 is described. Defining X, Xtilde, and Y to be the source alphabet, the compressed reproduction alphabet, and, the relevance variable respectively, it is demonstrated that minimization of a generalized IB (gIB) Lagrangian defined in terms of the nonadditivity parameter q* self-consistently yields the nonadditive effective distortion measure to be the q-deformed generalized Kullback-Leibler divergence: D
_K-L

^q
[p(Y|X)parp(Y|Xtilde)]. This result is achieved without enforcing any a-priori assumptions. Next, it is proven that the q*-deformed nonadditive free energy of the system is non-negative and convex. Finally, the update equations for the gIB method are derived. These results generalize critical features of the IB method to the case of Tsallis statistics.