Finite Partially Exchangeable Laws are Signed Mixtures of Product Laws

TL;DR

This paper characterizes finite partially exchangeable laws as signed mixtures of product laws, providing conditions for uniqueness and nonnegativity of the directing measures, with implications for exchangeability and reinforcement.

Contribution

It introduces a representation of finite partially exchangeable laws as signed mixtures of i.i.d. laws and establishes conditions for uniqueness and nonnegativity of the measures.

Findings

Law is a signed mixture of independent laws within classes

Uniqueness of representation linked to weak compactness of measures

Nonnegativity of the directing measure characterized by positive semi-definiteness

Abstract

Given a partition $\{I_1,\ldots,I_k\}$ of $\{1,\ldots,n\}$, let $(X_1,\ldots,X_n)$ be random vector with each $X_i$ taking values in an arbitrary measurable space $(S,\mathscr{S})$ such that their joint law is invariant under finite permutations of the indexes within each class $I_j$. Then, it is shown that this law has to be a signed mixture of independent laws and identically distributed within each class $I_j$. The representation is unique if and only if the set of these signed measures is weakly compact. We provide a necessary condition for the existence of a nonnegative directing measure. This is related to the notions of infinite extendibility and reinforcement. In the special case where $(X_1,\ldots,X_n)$ is an exchangeable sequence of $\{0,1\}$-valued random variables, the directing measure can be chosen nonnegative if and only if two effectively computable matrices are positive semi-definite.