An Information-Theoretic View of Stochastic Localization

Given a probability measure $\mu $ over notation="LaTeX">$\mathbb {R}^{n}$ , it is often useful to approximate by the convex combination of small number measures, such that each component close product measure. Recently, Ronen Eldan used stochastic localization argument prove general decomposition result this type. In Eldan’s theorem, ‘number components’ characterized entropy mixture, and ‘closeness product’ covariance matrix component. We present an elementary proof theorem which makes use information theory (or estimation theory) interpretation. The analogous one earlier known as ‘pinning lemma.’