[Paper Review] On the extendibility of finitely exchangeable probability measures

Takis Konstantopoulos, Linglong Yuan|arXiv (Cornell University)|Jan 25, 2015

Bayesian Methods and Mixture Models18 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition for the finite and infinite extendibility of finitely exchangeable probability measures on locally compact Hausdorff spaces using functional-analytic and measure-theoretic methods. The key contribution is a characterization of extendibility via bounded linear functionals and the Hahn-Banach theorem, generalizing de Finetti's theorem to finite exchangeability with signed directing measures.

ABSTRACT

A length-$n$ random sequence $X_1,\ldots,X_n$ in a space $S$ is finitely exchangeable if its distribution is invariant under all $n!$ permutations of coordinates. Given $N > n$, we study the extendibility problem: when is it the case that there is a length-$N$ exchangeable random sequence $Y_1,\ldots, Y_N$ so that $(Y_1,\ldots,Y_n)$ has the same distribution as $(X_1,\ldots,X_n)$? In this paper, we give a necessary and sufficient condition so that, for given $n$ and $N$, the extendibility problem admits a solution. This is done by employing functional-analytic and measure-theoretic arguments that take into account the symmetry. We also address the problem of infinite extendibility. Our results are valid when $X_1$ has a regular distribution in a locally compact Hausdorff space $S$. We also revisit the problem of representation of the distribution of a finitely exchangeable sequence.

Motivation & Objective

• To determine when a finitely exchangeable probability measure on a locally compact Hausdorff space can be extended to a longer or infinite exchangeable sequence.
• To generalize de Finetti's theorem to finite exchangeability by characterizing the directing signed measures that represent such distributions.
• To resolve the extendibility problem for exchangeable sequences using symmetric linear functionals and measure-theoretic tools.
• To provide a functional-analytic framework for extending finite exchangeable laws via bounded linear functionals on symmetric function spaces.

Proposed method

• Formulates the extendibility problem as a symmetric linear functional extension problem on the space of bounded measurable functions on $S^n$.
• Introduces the concept of a $k$-extending functional that preserves symmetry and maps to the integral of the marginal distribution.
• Applies the Hahn-Banach theorem to extend a primitive functional defined on diagonal functions to the full space of symmetric functions.
• Uses the Riesz representation theorem to associate the extended functional with a probability measure on $S^N$ when the functional is positive and bounded.
• Employs set functions and signed measures on the space of probability measures $\mathscr{P}(S)$ to represent the distribution of $n$-exchangeable sequences.
• Establishes that extendibility is equivalent to the existence of a symmetric, bounded linear functional satisfying the extension condition.

Experimental results

Research questions

• RQ1 Under what conditions can an $n$-exchangeable probability measure on a locally compact Hausdorff space be extended to an $N$-exchangeable measure for $N > n$?
• RQ2 What characterizes the directing signed measures that represent finite exchangeable sequences, and when are they extendible?
• RQ3 Is there a functional-analytic criterion for infinite extendibility of a finitely exchangeable law?
• RQ4 Can the Hahn-Banach theorem be used to construct a consistent extension of a symmetric functional defined on diagonal functions to the full symmetric function space?
• RQ5 What role does the total variation of the directing signed measure play in determining extendibility?

Key findings

• A necessary and sufficient condition for $N$-extendibility of an $n$-exchangeable law is the existence of a symmetric, bounded linear functional on the space of bounded measurable functions that extends the primitive functional defined by the marginal distribution.
• The extendibility problem reduces to the existence of a positive, bounded linear functional on the symmetric function space that agrees with the marginal expectation on diagonal functions.
• For the case $n=2$, the paper constructs an explicit $2$-extending functional using the Hahn-Banach theorem and proves its norm is 1, ensuring consistency with the marginal integral.
• The Riesz representation theorem guarantees that the extended functional corresponds to a probability measure on $S^N$ if the functional is positive, thus yielding a valid $N$-exchangeable extension.
• The directing signed measure $\nu$ representing the $n$-exchangeable law is not unique in general, but extendibility depends on the existence of a compatible extension of the associated functional.
• The paper shows that infinite extendibility is equivalent to the existence of a consistent sequence of $N$-extending functionals for all $N > n$, and provides a criterion based on functional extension and positivity.