Consider two experts with a common worldview but access to different information who are each asked to assign a probability to the event \(A\). This can be modelled by an underlying probability space \((\Omega,\mathcal{F},\mathbb{P})\) containing the event \(A\), and two sub-\(\sigma\)-fields \(\mathcal{G}\) and \(\mathcal{H}\) of \(\mathcal{F}\) representing the differing information. Let \(X=\mathbb{P}(A|\mathcal{G})\) and \(Y=\mathbb{P}(A|\mathcal{H})\), and define \[ \lambda(\delta)=\sup\mathbb{P}(|X-Y|>1-\delta)\,, \] where the supremum is taken over all probability spaces, events, and sub-\(\sigma\)-fields. For small \(\delta\), this can be interpreted as (a bound on) the probability that the two experts give very different estimates of the probability of \(A\).
The main result of the present paper is that \(\lambda(\delta)=\frac{2\delta}{1+\delta}\) for all \(0<\delta<\frac{1}{2}\). That this is a lower bound on \(\lambda(\delta)\) is a known result due to J. Pitman. In this paper, the authors give a proof that this is also an upper bound.