The final form of Tao's inequality relating conditional expectation and conditional mutual information

Summary form only given: Recently Terence Tao approached Szemeredi's regularity lemma from the perspectives of probability theory and of information theory instead of graph theory and found a stronger variant of this lemma, which involves a new parameter. To pass from an entropy formulation to an expectation formulation he found the following lemma. Let Y, and X, X' be discrete random variables taking values in y and x, respectively, where y sub [-1, 1], and with X' = f(X) for a (deterministic) function f. Then we have E(|E(Y|X') - E(Y|X)|) les 2I(X nland Y|X')1/2. We show that the constant 2 can be improved to (2ln2)1/2 and that this is the best possible constant