Loop Calculus and Belief Propagation for q-ary Alphabet: Loop Tower

Loop calculus introduced in (M. Chertkov and V. Chernyak, 2006) constitutes a new theoretical tool that explicitly expresses the symbol maximum-a-posteriori (MAP) solution of a general statistical inference problem via a solution of the belief propagation (BP) equations. This finding brought a new significance to the BP concept, which in the past was thought of as just a loop-free approximation. In this paper we continue a discussion of the loop calculus. We introduce an invariant formulation which allows to generalize the loop calculus approach to a q-are alphabet.