[Paper Review] Noncontinous additive entropies of partitions

Tomasz Sobieszek|arXiv (Cornell University)|Feb 21, 2012

Functional Equations Stability Results1 references3 citations

TL;DR

This paper extends the characterization of additive partition entropies by removing the continuity assumption, proving that any such entropy decomposes into a standard entropy and a component derived from a finitely additive set function into endomorphisms of the rationals. The key contribution is a homological result showing symmetric 2-cocycles on convex cones are coboundaries, resolving a functional equation central to the entropy decomposition.

ABSTRACT

In a previous paper: A. Paszkiewicz, T. Sobieszek, Additive Entropies of Partitions, we have given a description of additive partition entropies that is real functions $I$ on the set of finite partitions that are additive on stochastically independent partitions in a given probability space. We now present an analogical result, this time without assuming continuity. As a by-product of our efforts we solve a 2-cocycle functional equation for certain subsets of convex cones.

Motivation & Objective

• To characterize additive partition entropies without assuming continuity, extending prior results that required continuity.
• To solve a 2-cocycle functional equation for symmetric functions on convex cones.
• To prove that all symmetric solutions to the 2-cocycle equation on certain subsets of convex cones are coboundaries.
• To establish a structural decomposition of additive partition entropies into a standard entropy and a correction term via a finitely additive set function.

Proposed method

• Define a mapping ∆(V, W) ∈ End(R) for sets of equal P-measure, satisfying additivity and invariance under symmetric difference null sets.
• Construct a finitely additive set function m: X → End(R) vanishing on P-null sets such that ∆(V, W) = m(W) − m(V).
• Introduce a function f on convex cones that satisfies permutation symmetry and a generalized associativity condition (equation 3).
• Prove the existence of a function h: M → L such that f(a₁,…,aₙ) = h(∑aᵢ) − ∑h(aᵢ), using a Zorn’s lemma argument on the family of partial solutions.
• Use a ray-based construction on Q-linear spans to extend local solutions to the functional equation (7) h(a + b) = h(a) + h(b) + f(a, b) across convex cones.
• Apply the result to show that the entropy I decomposes as I = H_P + L_m, where H_P is a standard entropy and L_m is a logarithmic sum over m(Aᵢ) applied to log(1/P(Aᵢ)).

Experimental results

Research questions

• RQ1 Can additive partition entropies be fully characterized without assuming continuity, as in previous work?
• RQ2 Under what conditions is a symmetric 2-cocycle on a convex cone a coboundary?
• RQ3 Does every solution to the 2-cocycle equation on a convex cone with specified properties arise from a single function h via the coboundary formula?
• RQ4 What is the structural decomposition of an additive partition entropy when continuity is not assumed?

Key findings

• Any additive partition entropy I without continuity assumption decomposes as I = H_P + L_m, where H_P is a standard entropy and L_m is derived from a finitely additive set function m: X → End(R) vanishing on P-null sets.
• All symmetric solutions to the 2-cocycle functional equation on certain subsets of convex cones are coboundaries, as proven via a Zorn’s lemma argument on the family of partial solutions.
• The existence of a function h: M → L satisfying f(a₁,…,aₙ) = h(∑aᵢ) − ∑h(aᵢ) is guaranteed under the given symmetry and associativity conditions on f.
• The function f, which arises from the difference of m-values in the entropy decomposition, satisfies a generalized associativity and permutation symmetry, ensuring consistency across partitions.
• The construction of h is achieved by extending local solutions on rays and convex cones, using the condition that a + b ∈ M implies a, b ∈ M for the generating set M.
• The result implies that the L_m component of the entropy depends only on the measures of atoms if and only if m(A) = α(P(A)) for some fixed α ∈ End(R).