the conditional Jensen inequality

Ordinary Jensen's inequality says that for a convex function — one that bows upward, like a smile or a valley — the function of an average is at most the average of the function: phi(E[X]) <= E[phi(X)]. The conditional version simply replaces every plain expectation by a conditional one, and the same inequality survives intact, now holding as a relation between random variables rather than numbers.

Precisely: if phi is convex and X and phi(X) are integrable, then phi( E[X given G] ) <= E[ phi(X) given G ], almost surely. The picture is identical to the unconditional case: averaging X first and then bending it through the convex curve undershoots, because convexity means the chord lies above the curve — spreading the values out and then bending always lifts the average above the bent average. Conditioning on G is just doing this averaging within each block of information, so the inequality holds block by block, i.e. as an inequality between the two random variables. Taking phi(x) = x^2 gives the useful corollary ( E[X given G] )^2 <= E[ X^2 given G ], the seed of conditional variance being nonnegative.

Conditional Jensen is a quiet backbone of advanced probability. It proves that a convex function of a martingale is a submartingale (the engine behind Doob's inequalities), and it underlies contraction properties of conditioning in L^p spaces. The same caveat as the ordinary version: the direction depends entirely on convexity. For a concave function (bowing downward, like a hill) the inequality flips to phi(E[X given G]) >= E[phi(X) given G]; for a linear function it is an equality. Get the curvature wrong and you get the sign backwards.