Abstract
We propose that entropy is a universal co-homological class in a theory associated to a family of observable quantities and a family of probability distributions. Three cases are presented: (1) classical probabilities and random variables; (2) quantum probabilities and observable operators; (3) dynamic probabilities and observation trees. This gives rise to a new kind of topology for information processes, that accounts for the main information functions: entropy, mutual-informations at all orders, and Kullback–Leibler divergence and generalizes them in several ways. The article is divided into two parts, that can be read independently. In the first part, the introduction, we provide an overview of the results, some open questions, future results and lines of research, and discuss briefly the application to complex data. In the second part we give the complete definitions and proofs of the theorems A, C and E in the introduction, which show why entropy is the first homological invariant of a structure of information in four contexts: static classical or quantum probability, dynamics of classical or quantum strategies of observation of a finite system.
Highlights
Three cases are presented: (1) classical probabilities and random variables; (2) quantum probabilities and observable operators; (3) dynamic probabilities and observation trees. This gives rise to a new kind of topology for information processes, that accounts for the main information functions: entropy, mutual-informations at all orders, and Kullback–Leibler divergence and generalizes them in several ways
We suggest that all information quantities are of co-homological nature, in a setting which depends on a pair of categories; one for the data on a system, like random variables or functions of solutions of an equation, and one for the parameters of this system, like probability laws or coefficients of equations; the first category generates an algebraic structure like a monoid, or more generally a monad, and the second category generates a representation of this structure, as do for instance conditioning, or adding new numbers; information quantities are co-cycles associated with this module
We call random variables (r.v) on a finite set Ω congruent when they define the same partition (remind that a partition of Ω is a family of disjoint non-empty subsets covering Ω and that the partition associated to a r.v X is the family of subsets Ωx of Ω defined by the equations X(ω) = x); the join r.v Y Z, denoted by (Y, Z), corresponds to the less fine partition that is finer than Y and Z
Summary
It corresponds to a standard non-homogeneous bar complex (cf [5]) Another co-boundary operator on CN is δt (t for twisted or trivial action or topological complex), that is defined by the above formula with the first term S0.F (S1; ...; SN ; P) replaced by F (S1; ...; SN ; P). (cf Theorem 1 section 2.3, [7]): For the full simplex ∆(Ω), and if S is the monoid generated by a set of at least two variables, such that each pair takes at least four values, the information co-homology space of degree one is one-dimensional and generated by the classical entropy. The absolute minima of I3 correspond to Borromean links, interpreted as synergy, cf. [11,12]
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