Coherent distributions: Hilbert space approach and duality

Abstract

Let $X$ be a Bernoulli random variable with the success probability $p$. We are interested in tight bounds on $\mathbb{E}[f(X_1,X_2)]$, where $X_i=\mathbb{E}[X|{\mathcal{F}}_i]$ and ${\mathcal{F}}_i$ are some sigma-algebras. This problem is closely related to understanding extreme points of the set of coherent distributions. A distribution on $[0,1{]}^2$ is called $\text{coherent}$ if it can be obtained as the joint distribution of $(X_1,X_2)$ for some choice of ${\mathcal{F}}_i$. By treating random variables as vectors in a Hilbert space, we establish an upper bound for quadratic $f$, characterize $f$ for which this bound is tight, and show that such $f$ result in exposed coherent distributions with arbitrarily large support. As a corollary, we get a tight bound on $\mathrm{cov}(X_1,X_2)$ for $p\in [1/3,2/3]$. To obtain a tight bound on $\mathrm{cov}(X_1,X_2)$ for all $p$, we develop an approach based on linear programming duality. Its generality is illustrated by tight bounds on $\mathbb{E}[|X_1-X_2{|}^{\hat{I}}\pm ]$ for any $\hat{I}\pm >0$ and $p=1/2$.