Homotopy probability theory I

Abstract

ABSTRACT. This is the first of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. In this paper, cumulants are proved to coincide with morphisms of homotopy algebras. The sequel paper outlines how the framework presented here can assist in the development of homotopy probability theory, allowing the principles of derived mathematics to participate in classical and noncommutative probability theory.

Homotopy probability theory II

“…In the case of not necessarily commutative homotopy probability theory, where a is not necessarily commutative and L ∞ is replaced throughout with A ∞ , the product a can be extended as a coderivation on T V = ⊕ ∞ n=1 V ⊗n . The coalgebra isomorphism a, whose components V ⊗n → V are repeated multiplication using a, is the exponential of the extended coderivation [3]. In the commutative world the map a : SV → SV is defined to be the coalgebra isomorphism whose components S n V → V are repeated multiplications using a, but the coalgebra isomorphism a is not the exponential of the coderivation lift of of a.…”

“…The failure of the expectation value to respect the products in the space of random variables and the complex numbers can be processed to give an infinite sequence of operations κ n : V ⊗n → C. This sequence of operations assembles into an A ∞ morphism between two trivial A ∞ algebras. The main proposition in [3] is that the A ∞ morphism obtained via this process coincides with the cumulants of the initial probability space.…”

“…This paper is the second of two papers that interprets and utilizes this link. In the prequel to this paper [3], the authors present a deformation theoretic framework for studying maps between algebras that do not respect structure. The framework for studying maps that do not respect structure applies to probability theory in the following way.…”

“…Also, to keep the abstract ideas presented here connected to simple examples, the product of random variables will be assumed to be commutative throughout. So, L ∞ algebras, rather than A ∞ algebras, are used and the requisite modifications and results contained in [3] are given in Section 2.…”

Symmetry, Probabiliy, Entropy: Synopsis of the Lecture at MAXENT 2014

“…Probabilistic/entropic interpretation of homology, which is kind of "dual" to "homological interpretation of entropy-like invariants" by Bennequin [7], and also by Drummond-Cole et al [8,9], is also possible for "coupled systems" [10] where particularly attractive ones are systems of moving disjoint balls in space where the configuration spaces of these systems support rich homology structures that are induced from the classifying spaces of (subgroups of) infinite symmetric groups S ∞=I [11], that is expanded/corrected in [12].…”