Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Jensen's inequality
Summary
In mathematics, Jensen's 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. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations.
Jensen's inequality generalizes the statement that the secant line of a convex function lies above the graph of the function, which is Jensen's inequality for two points: the secant line consists of weighted means of the convex function (for t ∈ [0,1]),
while the graph of the function is the convex function of the weighted means,
Thus, Jensen's inequality is
In the context of probability theory, it is generally stated in the following form: if X is a random variable and φ is a convex function, then
The difference between the two sides of the inequality, , is called the Jensen gap.
The classical form of Jensen's inequality involves several numbers and weights. The inequality can be stated quite generally using either the language of measure theory or (equivalently) probability. In the probabilistic setting, the inequality can be further generalized to its full strength.
For a real convex function , numbers in its domain, and positive weights , Jensen's inequality can be stated as:
and the inequality is reversed if is concave, which is
Equality holds if and only if or is linear on a domain containing .
As a particular case, if the weights are all equal, then () and () become
For instance, the function log(x) is concave, so substituting in the previous formula () establishes the (logarithm of the) familiar arithmetic-mean/geometric-mean inequality:
A common application has x as a function of another variable (or set of variables) t, that is, .
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.