Nproof of jensen's inequality pdf

Now consider a probability distribution p on a set m and a function x assigning real values xm for m. First, we provide separate proofs for continuous functions followed by a general proof for all l 1 0, 1 functions. So for the case of the square function, jensen s inequality did not tell us anything that we didnt know. Jensens inequality in detail and sconvex functions 97 cannot state that this is the same as a generic situation, where the coe. R be a convex function, where 1 a theorem 12 jensens inequality if gis convex, then egx gex. Example from jensens inequality we see that ex2 ex2. Browse other questions tagged inequality convexanalysis contestmath proof explanation functional inequalities or ask your own question. Since the graph of every convex function lies above its tangent line at every point, we can compare the function with the linear function, whose graph is tangent to the graph of at the point. Theorem 1 jensen s inequality if f is convex then for any distribution p on m we have the following. The proof of jensens inequa lity does not address the specification of the cases of equality. Interpret jensen s inequality for the case in which the unit measure d.

Jensens inequality is of particular importance in statistical physics when the convex function is an exponential, giving. Convexity, inequalities, and norms 5 theorem 4 jensen s inequality integral version let x. Jensens inequality plays a central role in the derivation of the expectation maximization algorithm 1 and the proof of consistency of maximum likelihood estimators. Jensen s inequality tells us that e of g of x, with g the quadratic function, is larger than or equal to the square that is, g of the expected value. Browse other questions tagged probability inequality proof explanation convexanalysis jensen inequality or ask your own question. Proof we proceed by induction on n, the number of weights. Let us suppose, inductively, that jensens inequality holds for n. It can be shown that strict inequality exists unless all of the are equal or is linear on an interval containing all of the. Jensens inequality an overview sciencedirect topics.