Jensen's inequality proof pdf

The natural logarithm is a strictly concave function, because its second derivative is strictly negative on its domain of definition. Remember that variance of every random variable x is a positive value, i. Let us suppose, inductively, that jensen s inequality holds for n k 1. It is also a natural problem to give analogous results for the classical jensens inequality 1. It is also used to make claims about a function where little is known or needs to be known about the distribution. A refinement of the integral form of jensens inequality. Jensens inequality plays a central role in the derivation of the expectation maximization algorithm 1 and the proof of consistency of maximum likelihood estimators. This is a special case of the \principalagent problem of economics.

Qualitative predictions can be derived from the form of the relevant response. If we use the generalized jensens inequality instead, we can get the weighted amgm inequality. Browse other questions tagged inequality convexanalysis contestmath proofexplanation functionalinequalities or ask your own question. Replacing a convex function ux by a concave function cx merely reverses the inequality after y0. If n 1 then equality holds and the inequality is trivially true. A good explanation of jensens inequality should get the oh, duh reaction. What is an intuitive explanation of jensens inequality. I if the segment between any two points taken on its graph. Jensens inequality becomes equality only when n 1 or function u is affinelinear over at least the convex hull of the given arguments xj. Jensens inequality an overview sciencedirect topics. Pedros hopes and dreams i pedro is considering two ways to invest his life savings. Strong law of large numbers and jensens inequality scott she eld mit. This special kind of induction is done by performing the following steps. Overview of jensens inequality overview of the kantorovich inequality mondpecari.

In this article, ill try to provide such an explanation for the cauchyschwarz inequality, markovs inequality, and jensens inequality. For example, it is often used as a tool in mathematical proofs. Principle of mathematical induction inequality proof video duration. This is an extension of the definition of convexity on a finite number of points. The applications of functional variants of jensens inequality. Lecture notes 2 1 probability inequalities inequalities are useful for bounding quantities that might otherwise be hard to compute. Jensens inequality jensens inequality applies to convex functions. Jensen s inequality convex functions and a proof for. Jensens inequality is a mathematical property of nonlinear functions. There are many, many applications, but one i quite like is an alternative proof of the amgm inequality.

Jensens inequality predicts effects of environmental. The paper is inspired by mcshanes results on the functional form of jensens inequality for convex functions of several variables. We prove jensens inequality by an inductive argument on the number of points. Thus, we derive that g m belongs to an eigenspace of k and is also an eigenvector of k. The function gx x2 is an example of convex function. In mathematics, jensens inequality, named after the danish mathematician johan jensen, relates the value of a convex function of an integral to the integral of the convex function. This inequality is an equality if and only if for all m 10. The inequality states that for a nonlinear function. It can be shown that strict inequality exists unless all of the are equal or is linear on an interval containing all of the. Here we present the well known cauchys proof by induction. Inequality of arithmetic and geometric means wikipedia. Because of jensen s inequality, the convexity of the payo function is a genuine concern for hedge fund investors. For instance, x x k is the payoff function for an american call.

If its that obvious, itll be easy to extrapolate to other related circumstances. One simplification is to lower bound via jensens inequality or em. Jensens inequality is a useful tool in mathematics, specifically in applied fields such as probability and statistics. Jensen com putes simple lower bounds on otherwise intractable quantities such as products of sums and latent loglikelihoods. The proof of jensens inequality does not address the specification of the cases of equality. Intuitively a function is convex if it is upward bending. Jensens inequality is an inequality involving convexity of a function. Jensens inequality theorem for any concave function f, efx fex proof. Yao xie, ece587, information theory, duke university. People worry that it encourages fund managers like pedro to take risks that are bad for the client. Jensens inequality states that, for any convex function g, we have egx.

Hence, riskaversionisequivalent to the concavity of a bernoulli utility function ux. In spite of this, few papers have been published dealing with this problem see and. Here p n is the statement that the amgm is true for n variables. Suppose a strictly positive random variable has expected value and it is not constant with probability one. Jensens inequality is one of the most basic problem solving tools. As an example, we have generalized youngs inequality. Jensens inequality is a powerful mathematical tool and one of the workhorses in statistical learning. Section 3 is divided in two subsections, depending on the preorder relation and the assumptions on the convex functions. 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. Intuition behind the cauchyschwarz, markovs, and jensen.

A visual proof that p ab jensens and karamatas inequalities are related to the idea of smoothing, which allows you to make moves like jensen without necessarily the assumption of convexity. If we define gx x2, we can write the above inequality as egx. Proof we proceed by induction on n, the number of weights. Jensens inequality asserts there is an inequality associated to every convex function. Jensens 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. So for the case of the square function, jensens inequality did not tell us anything that we didnt know. For the sake of completness, we sketch the proof of andos theorem and give a short proof of jensens inequality for states in a c. Jensen s inequality plays a central role in the derivation of the expectation maximization algorithm 1 and the proof of consistency of maximum likelihood estimators. Strong law of large numbers and jensens inequality scott she eld mit 18.

Let us suppose, inductively, that jensens inequality holds for n k 1. What can we say about the expected value of, by using jensens inequality. Its applications therein include the em algorithm, bayesian estimation and bayesian inference. Since cx is strictly concave, this is possible only if all values. Given its generality, the inequality appears in many forms depending on the context. There is, however, usually an intuitive explanation of the inequality which allows you the understand what it actually means. If all w k 1, this reduces to the above inequality of arithmetic and geometric means. They will also be used in the theory of convergence. Instead of using the lagrange multiplier method or some other method based on differential calculus of several variables which might give complicated terms and equations difficult to handle, the utilitymaximization problems are solved by using jensens inequality and weighted arithmeticgeometric mean weighted amgm inequality. Consider the american derivative security with payoff g s k in period k. We show implementation and results of the bounds in applications i. Jensens inequality is of particular importance in statistical physics when the convex function is an exponential, giving. Jensens inequality wikimili, the best wikipedia reader.

Using the finite form of jensens inequality for the natural logarithm, we can prove the inequality between the weighted arithmetic mean and the weighted geometric mean stated above. Jensen 18591925, it was first described at the end of the 19th century6, 7. Jensen s inequality theorem for any concave function f, efx fex proof. Jensens inequality is a mathematical proof that is seldom mentioned in the ecological literature but which provides a powerful tool for predicting some direct effects of environmental variance in biological systems.