A Bayesian Proof of the Spread Lemma

A key set-theoretic “spread” lemma has been central to two recent celebrated results in combinatorics: the recent improvements on the sunflower conjecture by Alweiss, Lovett, Wu, and Zhang; and the proof of the fractional Kahn–Kalai conjecture by Frankston, Kahn, Narayanan, and Park. In this work, we present a new proof of the spread lemma, that—perhaps surprisingly—takes advantage of an explicit recasting of the proof in the language of Bayesian inference. We show that from this viewpoint the reasoning proceeds in a straightforward and principled probabilistic manner, leading to a truncated second moment calculation which concludes the proof.