Conditional Jensen inequality
= Conditional Jensen inequality
For an integrable <random variable> $X$, a sub-<sigma-algebra> $\mathcal G$, and a convex function $\phi$ with suitable integrability, $\phi(\mathbb E[X\mid\mathcal G])\le\mathbb E[\phi(X)\mid\mathcal G]$ almost surely. One proof writes $\phi$ as a supremum of supporting affine functions and applies monotonicity and <linearity> of <conditional expectation> to each. This proves the convex-loss form of the <Rao-Blackwell theorem>.