↧
Answer by Nawaf Bou-Rabee for Simple proof of Prékopa's Theorem:...
By the Brascamp–Lieb concentration inequality, we have $$ \operatorname{ Var}_z (D_{y}f) \le \langle (D_{zz}f)^{-1} (D_{zy} f)^2 \rangle_z \;,$$ and hence, $$g''(y) \ge \langle (D_{zz}f^{-1}) (D_{zz} f...
View ArticleSimple proof of Prékopa's Theorem: log-concavity is preserved by marginalization
The following result is well-known:Suppose that $H(x,y)$ is a log-concave distribution for $(x,y) \in \mathbb R^{m \times n}$ so that by definition we have$$H \left( (1 - \lambda)(x_1,y_1) + \lambda...
View Article