Jensen's Inequality

Link: https://home.ttic.edu/~dmcallester/ttic101-07/lectures/jensen/jensen.pdf

Jensen’s Inequality

Definition: A function f from the reals to the reals is convex if for every x1 and x2 and every p ∈ [0, 1] we have pf(x1) + (1 − p)f(x2) ≥ f(px1 + (1 − p)x2).

If f is (doubly) differentiable then f is convex if and only if df2/dx2 ≥ 0.

E[f(X)] >= f(E[x]) 函数的期望大于等于期望的函数

To apply this therome, the function must be convex: A twice-differentiable function g:I→ℝ is convex if and only if g″(x)≥0 for all x∈I.