**Jensen’s inequality** is a special inequality that has to do with convex sets. It says that if a particular function g is convex,

E[g(X)] ≥ g(E[X]).

Here E is the expected value, the mathematical expectation, or the average. It can be a probability weighted mean; so Jensen’s inequality also tells us that, if *w _{1}, w_{2}…w_{n}* are weights such that

Then, for arbitrary *x*:

If all the x_{j} are equal but one of the w_{j} equal zero, then the function f(x) the ≥ sign becomes an equals sign. If this is the only time in which that inequality becomes an inequality, we say our function is *strictly convex.*

## Defining Convex Functions

A **convex function** is where a line segment between any two points on the graph of a function is always above — or on—the graph.

## Applications of Jensen’s Inequality

Jensen’s Inequality is important in probability, in information theory, and in statistical physics.

In **probability**, it is central in the derivation of an important algorithm called the Expectation-Maximization algorithm. It also allows us to prove the consistency of maximum likelihood estimators. It can also be used to show that the arithmetic mean for a set of positive scalars is greater than or equal to their geometric mean. For how this is shown, see: Arithmetic Mean ≥ Geometric Mean.

In **statistical physics**, this inequality is most important if the convex function *g* is an exponential function, and where the expected values E are expected values with respect to a probability distribution.

In **information theory**, Jensen’s Inequality can be used to derive Gibbs inequality, which tells us about the mathematical entropy of discrete probability distributions.

## References

Smith, Andrew. Convex Sets and J’s Inequality. University College Dublin. Retrieved from https://www.ucd.ie/mathstat/t4media/convex-sets-and-jensen-inequalities-mathstat.pdf on July 20, 2019.

Derpanis, Konstantinos G. J’s Inequality. Version 1.0, March 2015. Retrieved from Retrieved from http://www.cs.yorku.ca/~kosta/CompVis_Notes/jensen.pdf on July 20, 2019

Pishro-Nik, Hossein. Introduction to Probability. Retrieved from https://www.probabilitycourse.com/chapter6/6_2_5_jensen%27s_inequality.php on July 20, 2019.

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