Towards Tsallis Fully Probabilistic Design

Abstract

Fully Probabilistic design (FPD) is a powerful framework offering an elegant and unifying account of stochastic control, learning and decision-making. Here we introduce a generalized FPD framework, which we term as Tsallis FPD. Tsallis FPD uses Tsallis divergence in place of the Kullback-Leibler divergence that defines the standard FPD cost term. Tsallis divergence is a natural generalization of the KL divergence, rooted in non-extensive statistical mechanics and providing flexibility towards modeling stochastic processes with non-Gaussian tail behavior. After formulating Tsallis FPD, we develop a constructive proof of convergence by formulating a fixed point iteration. The construction takes the form of a double iteration scheme that performs a sequence of backwards inductions, rather than a single pass down the stages that constitutes the proven approach for classical FPD. We prove that this construction asymptotically converges to a fixed point and that this fixed point is an optimal solution to Tsallis FPD.