Homotopy probability theory II
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Abstract
This is the second of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. This paper outlines how the framework can assist in the development of homotopy probability theory, where a vector space of random variables is replaced by a chain complex of random variables. This allows the principles of derived mathematics to participate in classical and noncommutative probability theory. A simple example is presented.
Keywords
Probability Cumulants HomotopyMathematics Subject Classification (2000)
55U35 46L53 60Axx1 Introduction
This paper is the second of two papers that interprets and utilizes a link between homotopy algebra and probability theory [9] found while studying certain algebraic aspects of quantum field theory [7, 8]. In the prequel [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. Expectation value is a linear map between a vector space \(V\) of random variables and the complex numbers that does not respect the product structure on \(V\). 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 \(\kappa _n:V^{\otimes n}\rightarrow \mathbb {C}\). This sequence of operations assembles into an \(A_\infty \) morphism between two trivial \(A_\infty \) algebras. The main proposition in [3] is that the \(A_\infty \) morphism obtained via this process coincides with the cumulants of the initial probability space.
In the case that the product of random variables is commutative, there is a similar construction of an infinite sequence of symmetric operations \(k_n:S^nV\rightarrow \mathbb {C}\) which assemble into an \(L_\infty \) morphism. In the commutative context, the result is that the \(L_\infty \) morphism coincides with the classical cumulants of the initial probability space.
The framework for studying maps which fail to preserve structure applies just as well when the space of random variables \(V\) is replaced by a chain complex \(C=(V,d)\). This generalization is pursued in this paper.

Classical and noncommutative probability theory are concerned with the expected value of a vector space of random variables equipped with an associative binary product.

Homotopy probability theory is concerned with the expected values of a chain complex of random variables equipped with an associative binary product.

Quantum field theory is concerned with the \(\mathbb {C}[[\hbar ]]\)valued expected values of a chain complex of random variables, called observables, equipped with an associative binary product satisfying certain subtle algebraic conditions related to \(\hbar \).
The reasons that quantum field theory frequently involves a differential have little to do with quantum fields. Rather, the differential is a tool used to handle symmetries by algebraic methods, and this tool can be applied to ordinary probability theory. In much the same way that one may resolve relations in homological algebra by replacing a module with a free resolution, one might extend a space of random variables with relations among expectation values to a larger chain complex of random variables where the relations are encoded in the differential. Then homotopical computations provide new ways to compute invariants of interest, such as joint moments. This idea is illustrated with a toy example in Sect. 4. Of course, there may be several ways for a given space of random variables to be extended to a chain complex of random variables, so care should be taken to understand which quantities are invariant of the original space of random variables and do not depend on choices made in a particular extension. This leads to the concept of homotopy random variables (Definition 7) and their moments and cumulants (Definitions 8 and 9).
Independent of any utilitarian advantages to replacing spaces of random variables by chain complexes of random variables, homotopy theoretic ideas provide attractive structural aspects to probability theory. Cumulants, for example, are interpreted as a homotopy morphism and then tools in homotopy theory, both computational and theoretical, can be brought to bear upon them. As another example, this point of view indicates how to construct a category of probability spaces that is different than previous constructions—there are, for example, many more morphisms. Moreover, once the framework for studying maps that do not respect structure is applied to probability theory, there become ways for the ideas, language, and tools of probability theory to participate in other areas that feature a nonstructure preserving map. Period integrals of smooth projective hypersurfaces are one such area where the participation seems to generate some new concepts [6].
The authors would like to thank Tyler Bryson, Joseph Hirsh, Tom LeGatta, and Bruno Vallette for many helpful discussions.
2 Commutative homotopy probability spaces and \(L_\infty \) algebras
Definition 1
Remark 1
Note that \(a\) is not required to possess any sort of compatibility with \(d\) or \(e\) (which it will not in interesting examples).
Remark 2
There are multiple easy generalizations to this definition that this paper will not pursue. It is natural to replace \(\mathbb {C}\) with a different ground algebra. It is also reasonable to allow \(a\) to be any element of \(\hom (SV,V)\). For example, \(a\) might be a nonassociative binary product, or a collection of nonbinary products.
Definition 2
A morphism of commutative homotopy probability spaces between \((C,e,a)\) and \((C',e',a')\) is a chain map \(f:C\rightarrow C'\) that commutes with expectation in the sense that \(e'f=e\).
Remark 3
Note again that there is no compatibility assumed between \(f\) and the products \(a\) and \(a'\).
Basic facts and definitions about \(L_\infty \) algebras are now recalled. For more details, see [4]. Let \(V\) be a graded vector space. Let \(S^n V\) be the \(S_n\)invariant subspace of \(V^{\otimes n}\) and let \(SV=\oplus _{n=1}^\infty S^nV\). As a direct sum, linear maps from \(SV\) to a vector space \(W\) correspond to collections of linear maps \(\{S^nV \rightarrow W\}_{n=1}^\infty \). Also, \(SV\) is a coalgebra, free in a certain sense, so that linear maps from a commutative coalgebra \(\mathcal {C}\rightarrow V\) are in bijection with the following two sets: \(\{\)coalgebra maps from \(\mathcal {C}\) to \(SV \}\) and \(\{\)coderivations from \(\mathcal {C}\) to \(SV\}\).
Definition 3
An \(L_\infty \) algebra is a pair \((V,D)\) where \(V\) is a graded vector space and \(D:SV \rightarrow SV\) is a degree one coderivation satisfying \(D^2=0\). An \(L_\infty \) morphism between two \(L_\infty \) algebras \((V,D)\) and \((V',D')\) is a differential coalgebra map \(F:(SV,D)\rightarrow (SV',D')\). In other words, an \(L_\infty \) map from \((V,D)\) to \((V',D')\) is a degree zero coalgebra map \(F:SV \rightarrow SV'\) satisfying \(FD=D'F\).
Remark 4
Often, the definition of an \(L_\infty \) algebra involves a degree shift (so according to that convention, a degree one coderivation \(D:SV \rightarrow SV\) satisfying \(D^2=0\) would define an \(L_\infty \) algebra on the underlying vector space \(V[1]\)). For the applications to probability theory proposed in this paper, the conventional degree shift makes the signs much more complicated—it’s simpler to eliminate the shift in the definition.
Construction
Definition 4
Let \((V,D)\) and \((V',D')\) be \(L_\infty \) algebras. Let \(\Omega \) denote the commutative differential graded algebra \(\mathbb {C}[t,dt],\) polynomials in a variable \(t\) of degree \(0\) and its differential \(dt\) (so in particular \((dt)^2=0\)). An \(L_\infty \) homotopy \(H\) from \((V,D)\) to \((V',D')\) is an \(L_\infty \) morphism from \((V,D)\) to \((V'_\Omega ,D'_\Omega )\). Given an \(L_\infty \) homotopy \(H\), the evaluation maps \(\Omega \rightarrow \mathbb {C}\) for \(t=0\) and \(t=1\) induce two \(L_\infty \) morphisms from \((V,D)\) to \((V',D')\). Call these two \(L_\infty \) morphisms \(f\) and \(f'\). We say that \(H\) is a homotopy between \(f\) and \(f'\).
Remark 5
Homotopy is an equivalence relation on \(L_\infty \) morphisms from \((V,D)\) to \((V',D')\). Transitivity is not obvious but can be verified [5].
Remark 6
Homotopy behaves well with respect to composition. That is, if \(f\) and \(f'\) are homotopic \(L_\infty \) maps \(P\rightarrow Q\) and \(g\) and \(g'\) are homotopic \(L_\infty \) maps \(Q\rightarrow R\), then \(g\circ f\) and \(g'\circ f'\) are homotopic \(L_\infty \) maps \(P\rightarrow R\). In particular, this is true in the special cases where \(f=f'\) or when \(g=g'\).
Lemma 1
Let \((V,0)\) be a trivial \(L_\infty \) algebra. If \(f\) and \(f'\) are homotopic \(L_\infty \) morphisms \((V,0)\rightarrow (\mathbb {C},0)\), then \(f=f'\).
Proof
Let \(H\) be a homotopy between \(f\) and \(f'\). Then \(H\) is an \(L_\infty \) morphism from \((V,0)\) to \(\Omega \). That is, \(H\) is a coalgebra map \(SV\rightarrow S\Omega \) satisfying \(dH=0\). In particular, each of the constituent maps \(S^nV\rightarrow \Omega \) must have closed image in \(\Omega \), so they must actually land in \(\mathbb {C}\oplus dt\mathbb {C}[t,dt]\). This means that the evaluation of \(H\) at \(0\) and at \(1\) yield the same map to \(\mathbb {C}\), so \(f\) and \(f'\) coincide.\(\square \)
3 Expectation, moments, and cumulants
Notation
Let \((C,e,a)\) be a commutative homotopy probability space, with \(C=(V,d)\). Then \((V,d)\) and \((\mathbb {C},0)\) are \(L_\infty \) algebras with vanishing higher maps and \(e\) is an \(L_\infty \) morphism from \((V,d)\) to \((\mathbb {C},0)\). Let \(\mathbf {a}\) denote the isomorphism of coalgebras \(SV\rightarrow SV\) whose \(n\)th component \(S^nV\rightarrow V\) is repeated multiplication using \(a\). Let \(\mathbf {a}'\) denote the isomorphism of coalgebras \(S\mathbb {C}\rightarrow S\mathbb {C}\) whose \(n\)th component \(S^n\mathbb {C}\rightarrow \mathbb {C}\) is repeated complex multiplication. The lowest component of \(\mathbf {a}\) and \(\mathbf {a}'\) are the identity.
Remark 7
In the case of not necessarily commutative homotopy probability theory, where \(a\) is not necessarily commutative and \(L_\infty \) is replaced throughout with \(A_\infty \), the product \(a\) can be extended as a coderivation on \(TV=\oplus _{n=1}^\infty V^{\otimes n}\). The coalgebra isomorphism \(\mathbf {a}\), whose components \(V^{\otimes n}\rightarrow V\) are repeated multiplication using \(a\), is the exponential of the extended coderivation [3]. In the commutative world the map \(\mathbf {a}:SV\rightarrow SV\) is defined to be the coalgebra isomorphism whose components \(S^nV\rightarrow V\) are repeated multiplications using \(a\), but the coalgebra isomorphism \(\mathbf {a}\) is not the exponential of the coderivation lift of of \(a\).
Definition 5
Definition 6
Moments and cumulants of a commutative homotopy probability space are homotopy invariant.
Proposition 1
Let \(C\) be a chain complex equipped with a product \(a\). Let \(e\) and \(e'\) be homotopic chain maps \(C\rightarrow (\mathbb {C},0)\). Then the total moment \(M\) of the commutative homotopy probability space \((C,e,a)\) is homotopic to the total moment \(M'\) of the commutative homotopy probability space \((C,e',a)\). Also, the total cumulant \(K\) of the commutative homotopy probability space \((C,e,a)\) is homotopic to the total cumulant \(K'\) of the commutative homotopy probability space \((C,e',a)\).
Proof
The transferred homotopy \(H^{\mathbf {a}, \mathrm{id }}\) is a homotopy between the total moments. The transferred homotopy \(H^{\mathbf {a},\mathbf {a}'}\) is a homotopy between the total cumulants.\(\square \)
In applications, one would like to evaluate moments and cumulants on random variables and obtain numbers. Direct application of moments and cumulants yield numbers that are not homotopy invariant. Homotopy random variables are now introduced—they have joint moments and cumulants that are homotopy invariant.
Definition 7
A collection of homotopy random variables \((X_1,\ldots , X_n)\) in a commutative homotopy probability space \((C,e,a)\) is the homotopy class of an \(L_\infty \) map \((\mathbb {C}^n,0)\rightarrow (V,D^\mathbf {a})\). The various factors of \(\mathbb {C}^n\) may be in different degrees. When \(n=1\), we call such a collection a homotopy random variable.
Example 1
The simplest kind of homotopy random variable is an \(L_\infty \) morphism \(X:(\mathbb {C},0)\rightarrow (V,D^\mathbf {a})\) whose only nonzero component is a map \(\mathbb {C}\rightarrow V\). Such a map is determined by \(1\mapsto x\) for an element \(x\in V\). Not any choice of element \(x\in V\) will define a homotopy random variable. The condition that \(X\) be an \(L_\infty \) morphism is that \(D^\mathbf {a}X=0\), which encodes an infinite collection of conditions on \(x\). Namely, \(dx=0\), \(d_2^\mathbf {a}(x,x)=0\), etc... More generally, a single homotopy random variable \(X\) will have components \(S^k\mathbb {C}\rightarrow V\), each of which is given by a map \(S^k1\mapsto x_k\) for some \(x_k\in V\). So, an arbitrary single random variable can be thought of as a sequence of elements \(\{x_k\in V\}\). Note that every closed element \(x\in V\) gives rise to a homotopy random variable by transport. That is, if \(x\in V\) satisfies \(dx=0\), then \(1\mapsto x\) defines a chain map \(f:(\mathbb {C},0)\rightarrow (V,d)\). Such a chain map is an \(L_\infty \) morphism \(F:(\mathbb {C},0)\rightarrow (V,D)\). The transport of \(F\) to an \(L_\infty \) map \(F^{\mathrm{id },\mathbf {a}}:(\mathbb {C},0)\rightarrow (V,D^\mathbf {a})\) is a homotopy random variable. Note that not all homotopy random variables arise as the transport of chain maps; an \(L_\infty \) morphism \(X:(\mathbb {C},0)\rightarrow (V,D^\mathbf {a})\) can be transported back to give \(L_\infty \) morphisms \(X^{\mathrm{id },(\mathbf {a})^{1}}:(\mathbb {C},0)\rightarrow (V,D)\), but this map may not only be a chain map, there may be nonzero components \(S^k\mathbb {C}\rightarrow V\) for \(k>1\).
Example 2
Remark 8
Given a collection of homotopy random variables \((X_1,\ldots , X_n)\), one can obtain individual homotopy random variables \(X_i\) by precomposing the inclusion of \(\mathbb {C}\) into \(\mathbb {C}^n\) in the \(i\)th factor. It is not true in general that given a set \(\{X_i\}_{i=1,\ldots , n}\) of random variables one can meaningfully generate a collection of homotopy random variables \((X_1,\ldots , X_n)\).
Definition 8
Definition 9
Lemma 2
Joint moments and joint cumulants are welldefined.
Proof
Let \(f\) and \(f'\) be two representatives of the homotopy class \((X_1,\ldots , X_n)\). Then \(M \circ f\) and \(M\circ f'\) are homotopic \(L_\infty \) maps from \((\mathbb {C}^n,0)\) to \((\mathbb {C},0)\). By Lemma 1, these two maps coincide. The compositions \(K \circ f\) and \(K\circ f'\) coincide for the same reason.\(\square \)
Remark 9
The \(L_\infty \) structure \(D^\mathbf {a}\) does not depend on \(e\). A collection of homotopy random variables could be defined for a pair \((C,a)\) without reference to an expectation. In particular, a collection of homotopy random variables \((X_1, \ldots , X_n)\) for a commutative homotopy probability space \((C,e,a)\) is also a collection of homotopy random variables for a commutative homotopy probability space \((C,e',a)\).
Proposition 2
Proof
Cumulants evaluated on a collection of homotopy random variables constitute \(L_\infty \) morphisms from \((\mathbb {C}^n,0)\rightarrow (\mathbb {C},0)\). The proposition shows that in the case in question, two such \(L_\infty \) maps are homotopic. By Lemma 1, they in fact coincide.\(\square \)
4 The Gaussian
If \(f=f(x)\) and \(g=g(x)\) are polynomials that represent the same homology class in \((V,d)\), then the maps \(1\mapsto f\) and \(1\mapsto g\) define homotopic chain maps \((\mathbb {C},0)\rightarrow (V,d)\). The corresponding probability theory statement is that their expectations agree \(e(f)=e(g)\). However, the moments of \(f\) and \(g\) will not agree in general, for example \(e(f^2)\) and \(e(g^2)\) need not agree. If, however, the maps \(1\mapsto f\) and \(1\mapsto g\) define homotopic \(L_\infty \) maps \((\mathbb {C},0)\rightarrow (V,D^\mathbf {a})\), then not only will their expectations agree, but all of their moments will agree as well. This can be verified explicitly.
The \(L_\infty \) algebra \((V,D^\mathbf {a})\) in this case is a differential graded Lie algebra, the bracket defined by \(d_2^\mathbf {a}\) may be familiar as the Poisson bracket defined on the polynomial functions defined on an odd symplectic vector space. Since both \(d(f)=0\) and \(d_2^\mathbf {a}(f,f)=0\) for a polynomial \(f=f(x)\), the map \(1\mapsto f\) is an \(L_\infty \) map \((\mathbb {C},0)\rightarrow (V,D^\mathbf {a})\). Because joint moments of homotopy random variables are well defined, if two maps \(1\mapsto f\) and \(1\mapsto g\) are homotopic \(L_\infty \) maps then \(e(f^n)=e(g^n)\) for all \(n\).
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