  This note concerns a well-known result which we term the ``spread lemma,''
which establishes the existence (with high probability) of a desired structure
in a random set. The spread lemma was central to two recent celebrated results:
(a) the improved bounds of Alweiss, Lovett, Wu, and Zhang (2019) on the
Erd\H{o}s-Rado sunflower conjecture; and (b) the proof of the fractional
Kahn--Kalai conjecture by Frankston, Kahn, Narayanan and Park (2019). While the
lemma was first proved (and later refined) by delicate counting arguments,
alternative proofs have also been given, via Shannon's noiseless coding theorem
(Rao, 2019), and also via manipulations of Shannon entropy bounds (Tao, 2020).
  In this note we present a new proof of the spread lemma, that takes advantage
of an explicit recasting of the proof in the language of Bayesian statistical
inference. We show that from this viewpoint the proof proceeds in a
straightforward and principled probabilistic manner, leading to a truncated
second moment calculation which concludes the proof. The proof can also be
viewed as a demonstration of the ``planting trick'' introduced by Achlioptas
and Coga-Oghlan (2008) in the study of random constraint satisfaction problems.
