A homological characterization of generalized multinomial coefficients related to the entropic chain rule

Abstract

There is an asymptotic relationship between the multiplicative relations among multinomial coefficients and the (additive) recurrence property of Shannon entropy known as the chain rule. We show that both types of identities are manifestations of a unique algebraic construction: a $1$-cocycle condition in \emph{information cohomology}, an algebraic invariant of phesheaves of modules on \emph{information structures} (categories of observables). Baudot and Bennequin introduced this cohomology and proved that Shannon entropy represents the only nontrivial cohomology class in degree $1$ when the coefficients are a natural presheaf of probabilistic functionals. The author obtained later a $1$-parameter family of deformations of that presheaf, in such a way that each Tsallis $α$-entropy appears as the unique $1$-cocycle associated to the parameter $α$. In this article, we introduce a new presheaf of \emph{combinatorial functionals}, which are measurable functions of finite arrays of integers; these arrays represent \emph{histograms} associated to random experiments. In this case, the only cohomology class in degree $0$ is generated by the exponential function and $1$-cocycles are Fontené-Ward generalized multinomial coefficients. As a byproduct, we get a simple combinatorial analogue of the fundamental equation of information theory that characterizes the generalized binomial coefficients. The asymptotic relationship mentioned above is extended to a correspondence between certain generalized multinomial coefficients and any $α$-entropy, that sheds new light on the meaning of the chain rule and its deformations.