Jensen's Inequality

Probability · Axiom Academy

Averaging before you apply a function and averaging after are not the same thing — and curvature decides which one wins. 1. Convex Means the Chord Sits Above Everything here rests on one geometric property. A function g is convex on an interval if, for every pair of points you pick on its graph, the straight chord joining them lies on or above the curve between them. Watch the endpoints slide apart below — the chord never once dips beneath the arc. Written algebraically, "the chord is above the curve" says that for any two points a and b and any weight between 0 and 1: Convex: the weighted average of the heights is at least the height at the weighted average The point slides along the x -axis between a and b as runs from 1 to 0. The left side of the inequality is the height of the chord above that point; the right side is the height of the curve . A function is concave when this is reversed — when the chord lies on or below the curve. x^2 , |x| , e^ x , 1/x on x>0 . A twice-differentiable g is convex exactly when . and on x>0 . Here — the curve bends the other way. A straight line g(x)=mx+k is convex and concave: the chord lies exactly on the curve. over a full period curves up in places and down in others, so no single direction applies. 2. Averaging Inside vs. Averaging Outside

This is the written version of the interactive lesson above. See the full Probability course.