Jensen's inequality

Proofs

Intuitive graphical proof

Jensen's inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be offered. Before embarking on these mathematical derivations, however, it is worth analyzing an intuitive graphical argument based on the probabilistic case where X is a real number (see figure). Assuming a hypothetical distribution of X values, one can immediately identify the position of ${\displaystyle \mathrm{E}[X]}$ and its image ${\displaystyle \phi (\mathrm{E}[X])}$ in the graph. Noticing that for convex mappings Y = φ(x) of some x values the corresponding distribution of Y values is increasingly "stretched up" for increasing values of X, it is easy to see that the distribution of Y is broader in the interval corresponding to X > X₀ and narrower in X < X₀ for any X₀; in particular, this is also true for ${\displaystyle X_0=\mathrm{E}[X]}$. Consequently, in this picture the expectation of Y will always shift upwards with respect to the position of ${\displaystyle \phi (\mathrm{E}[X])}$. A similar reasoning holds if the distribution of X covers a decreasing portion of the convex function, or both a decreasing and an increasing portion of it. This "proves" the inequality, i.e.

${\displaystyle \phi (\mathrm{E}[X])\le \mathrm{E}[\phi (X)]=\mathrm{E}[Y],}$

with equality when φ(X) is not strictly convex, e.g. when it is a straight line, or when X follows a degenerate distribution (i.e. is a constant).

The proofs below formalize this intuitive notion.

Proof 1 (finite form)

If λ₁ and λ₂ are two arbitrary nonnegative real numbers such that λ₁ + λ₂ = 1 then convexity of φ implies

${\displaystyle \mathrm{\forall }{x}_1,x_2:\phi \left({\lambda }_1{x}_1+{\lambda }_2{x}_2\right)\le {\lambda }_1\phi (x_1)+{\lambda }_2\phi (x_2).}$

This can be generalized: if λ₁, ..., λ_n are nonnegative real numbers such that λ₁ + ... + λ_n = 1, then

${\displaystyle \phi ({\lambda }_1{x}_1+{\lambda }_2{x}_2+\cdots +{\lambda }_n{x}_n)\le {\lambda }_1\phi (x_1)+{\lambda }_2\phi (x_2)+\cdots +{\lambda }_n\phi (x_n),}$

for any x₁, ..., x_n.

The finite form of the Jensen's inequality can be proved by induction: by convexity hypotheses, the statement is true for n = 2. Suppose the statement is true for some n, so

${\displaystyle \phi \left(\sum _{i=1}^n{\lambda }_i{x}_i\right)\le \sum _{i=1}^n{\lambda }_i\phi \left(x_i\right)}$

for any λ₁, ..., λ_n such that λ₁ + ... + λ_n = 1.

One needs to prove it for n + 1. At least one of the λ_i is strictly smaller than ${\displaystyle 1}$, say λ_n+1; therefore by convexity inequality:

${\displaystyle \begin{array}{cc}\phi \left(\sum _{i=1}^{n+1}{\lambda }_i{x}_i\right)& =\phi \left((1-{\lambda }_{n+1})\sum _{i=1}^n\frac{{\lambda }_i}{1-{\lambda }_{n+1}}{x}_i+{\lambda }_{n+1}{x}_{n+1}\right)\\ & \le (1-{\lambda }_{n+1})\phi \left(\sum _{i=1}^n\frac{{\lambda }_i}{1-{\lambda }_{n+1}}{x}_i\right)+{\lambda }_{n+1}\phi (x_{n+1}).\end{array}}$

Since λ₁ + ... +λ_n + λ_n+1 = 1,

${\displaystyle \sum _{i=1}^n\frac{{\lambda }_i}{1-{\lambda }_{n+1}}=1}$,

applying the inductive hypothesis gives

${\displaystyle \phi \left(\sum _{i=1}^n\frac{{\lambda }_i}{1-{\lambda }_{n+1}}{x}_i\right)\le \sum _{i=1}^n\frac{{\lambda }_i}{1-{\lambda }_{n+1}}\phi (x_i)}$

therefore

${\displaystyle \begin{array}{cc}\phi \left(\sum _{i=1}^{n+1}{\lambda }_i{x}_i\right)& \le (1-{\lambda }_{n+1})\sum _{i=1}^n\frac{{\lambda }_i}{1-{\lambda }_{n+1}}\phi (x_i)+{\lambda }_{n+1}\phi (x_{n+1})=\sum _{i=1}^{n+1}{\lambda }_i\phi (x_i)\end{array}}$

We deduce the inequality is true for n + 1, by induction it follows that the result is also true for all integer n greater than 2.

In order to obtain the general inequality from this finite form, one needs to use a density argument. The finite form can be rewritten as:

${\displaystyle \phi \left(\int xd{\mu }_n(x)\right)\le \int \phi (x)d{\mu }_n(x),}$

where μ_n is a measure given by an arbitrary convex combination of Dirac deltas:

${\displaystyle {\mu }_n=\sum _{i=1}^n{\lambda }_i{\delta }_{x_i}.}$

Since convex functions are continuous, and since convex combinations of Dirac deltas are weakly dense in the set of probability measures (as could be easily verified), the general statement is obtained simply by a limiting procedure.

Proof 2 (measure-theoretic form)

Let ${\displaystyle g}$ be a real-valued ${\displaystyle \mu }$-integrable function on a probability space ${\displaystyle \mathrm{\Omega }}$, and let ${\displaystyle \phi }$ be a convex function on the real numbers. Since ${\displaystyle \phi }$ is convex, at each real number ${\displaystyle x}$ we have a nonempty set of subderivatives, which may be thought of as lines touching the graph of ${\displaystyle \phi }$ at ${\displaystyle x}$, but which are below the graph of ${\displaystyle \phi }$ at all points (support lines of the graph).

Now, if we define

${\displaystyle x_0:={\int }_{\mathrm{\Omega }}gd\mu ,}$

because of the existence of subderivatives for convex functions, we may choose ${\displaystyle a}$ and ${\displaystyle b}$ such that

${\displaystyle ax+b\le \phi (x),}$

for all real ${\displaystyle x}$ and

${\displaystyle a{x}_0+b=\phi (x_0).}$

But then we have that

${\displaystyle \phi \circ g(\omega )\ge ag(\omega )+b}$

for almost all ${\displaystyle \omega \in \mathrm{\Omega }}$. Since we have a probability measure, the integral is monotone with ${\displaystyle \mu (\mathrm{\Omega })=1}$ so that

${\displaystyle {\int }_{\mathrm{\Omega }}\phi \circ gd\mu \ge {\int }_{\mathrm{\Omega }}(ag+b)d\mu =a{\int }_{\mathrm{\Omega }}gd\mu +b{\int }_{\mathrm{\Omega }}d\mu =a{x}_0+b=\phi (x_0)=\phi \left({\int }_{\mathrm{\Omega }}gd\mu \right),}$

as desired.

Proof 3 (general inequality in a probabilistic setting)

Let X be an integrable random variable that takes values in a real topological vector space T. Since ${\displaystyle \phi :T\to \mathbb{R}}$ is convex, for any ${\displaystyle x,y\in T}$, the quantity

${\displaystyle \frac{\phi (x+\theta y)-\phi (x)}{\theta },}$

is decreasing as θ approaches 0⁺. In particular, the subdifferential of ${\displaystyle \phi }$ evaluated at x in the direction y is well-defined by

${\displaystyle (D\phi )(x)\cdot y:=\underset{\theta \downarrow 0}{lim}\frac{\phi (x+\theta y)-\phi (x)}{\theta }=\underset{\theta \ne 0}{inf}\frac{\phi (x+\theta y)-\phi (x)}{\theta }.}$

It is easily seen that the subdifferential is linear in y (that is false and the assertion requires Hahn-Banach theorem to be proved) and, since the infimum taken in the right-hand side of the previous formula is smaller than the value of the same term for θ = 1, one gets

${\displaystyle \phi (x)\le \phi (x+y)-(D\phi )(x)\cdot y.}$

In particular, for an arbitrary sub-σ-algebra ${\displaystyle \mathfrak{G}}$ we can evaluate the last inequality when ${\displaystyle x=\mathrm{E}[X\mid \mathfrak{G}],y=X-\mathrm{E}[X\mid \mathfrak{G}]}$ to obtain

${\displaystyle \phi (\mathrm{E}[X\mid \mathfrak{G}])\le \phi (X)-(D\phi )(\mathrm{E}[X\mid \mathfrak{G}])\cdot (X-\mathrm{E}[X\mid \mathfrak{G}]).}$

Now, if we take the expectation conditioned to ${\displaystyle \mathfrak{G}}$ on both sides of the previous expression, we get the result since:

${\displaystyle \mathrm{E}\left[\left[(D\phi )(\mathrm{E}[X\mid \mathfrak{G}])\cdot (X-\mathrm{E}[X\mid \mathfrak{G}])\right]\mid \mathfrak{G}\right]=(D\phi )(\mathrm{E}[X\mid \mathfrak{G}])\cdot \mathrm{E}[\left(X-\mathrm{E}[X\mid \mathfrak{G}]\right)\mid \mathfrak{G}]=0,}$

by the linearity of the subdifferential in the y variable, and the following well-known property of the conditional expectation:

${\displaystyle \mathrm{E}\left[\left(\mathrm{E}[X\mid \mathfrak{G}]\right)\mid \mathfrak{G}\right]=\mathrm{E}[X\mid \mathfrak{G}].}$

Applications and special cases

Form involving a probability density function

Suppose Ω is a measurable subset of the real line and f(x) is a non-negative function such that

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=1.}$

In probabilistic language, f is a probability density function.

Then Jensen's inequality becomes the following statement about convex integrals:

If g is any real-valued measurable function and $\phi$ is convex over the range of g, then

${\displaystyle \phi \left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}g(x)f(x)dx\right)\le {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\phi (g(x))f(x)dx.}$

If g(x) = x, then this form of the inequality reduces to a commonly used special case:

${\displaystyle \phi \left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}xf(x)dx\right)\le {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\phi (x)f(x)dx.}$

This is applied in Variational Bayesian methods.

Example: even moments of a random variable

If g(x) = x²ⁿ, and X is a random variable, then g is convex as

${\displaystyle \frac{d^{2}g}{d{x}^2}(x)=2n(2n-1)x^{2n-2}\ge 0\mathrm{\forall }x\in \mathbb{R}}$

and so

${\displaystyle g(\mathrm{E}[X])=(\mathrm{E}[X]{)}^{2n}\le \mathrm{E}[X^{2n}].}$

In particular, if some even moment 2n of X is finite, X has a finite mean. An extension of this argument shows X has finite moments of every order ${\displaystyle l\in \mathbb{N}}$ dividing n.

Alternative finite form

Let Ω = {x₁, ... x_n}, and take μ to be the counting measure on Ω, then the general form reduces to a statement about sums:

${\displaystyle \phi \left(\sum _{i=1}^{n}g(x_i){\lambda }_i\right)\le \sum _{i=1}^n\phi (g(x_i)){\lambda }_i,}$

provided that λ_i ≥ 0 and

${\displaystyle {\lambda }_1+\cdots +{\lambda }_n=1.}$

There is also an infinite discrete form.

Statistical physics

Jensen's inequality is of particular importance in statistical physics when the convex function is an exponential, giving:

${\displaystyle e^{\mathrm{E}[X]}\le \mathrm{E}\left[e^X\right],}$

where the expected values are with respect to some probability distribution in the random variable X.

Information theory

If p(x) is the true probability density for X, and q(x) is another density, then applying Jensen's inequality for the random variable Y(X) = q(X)/p(X) and the convex function φ(y) = −log(y) gives

${\displaystyle \mathrm{E}[\phi (Y)]\ge \phi (\mathrm{E}[Y])}$

Therefore:

${\displaystyle -D(p(x)\Vert q(x))=\int p(x)\log \left(\frac{q(x)}{p(x)}\right)dx\le \log \left(\int p(x)\frac{q(x)}{p(x)}dx\right)=\log \left(\int q(x)dx\right)=0}$

a result called Gibbs' inequality.

It shows that the average message length is minimised when codes are assigned on the basis of the true probabilities p rather than any other distribution q. The quantity that is non-negative is called the Kullback–Leibler divergence of q from p, where ${\displaystyle D(p(x)\Vert q(x))=\int p(x)\log \left(\frac{p(x)}{q(x)}\right)dx}$.

Since −log(x) is a strictly convex function for x > 0, it follows that equality holds when p(x) equals q(x) almost everywhere.

Rao–Blackwell theorem

If L is a convex function and ${\displaystyle \mathfrak{G}}$ a sub-sigma-algebra, then, from the conditional version of Jensen's inequality, we get

${\displaystyle L(\mathrm{E}[\delta (X)\mid \mathfrak{G}])\le \mathrm{E}[L(\delta (X))\mid \mathfrak{G}]⟹\mathrm{E}[L(\mathrm{E}[\delta (X)\mid \mathfrak{G}])]\le \mathrm{E}[L(\delta (X))].}$

So if δ(X) is some estimator of an unobserved parameter θ given a vector of observables X; and if T(X) is a sufficient statistic for θ; then an improved estimator, in the sense of having a smaller expected loss L, can be obtained by calculating

${\displaystyle {\delta }_1(X)={\mathrm{E}}_{\theta }[\delta (X^{\prime })\mid T(X^{\prime })=T(X)],}$

the expected value of δ with respect to θ, taken over all possible vectors of observations X compatible with the same value of T(X) as that observed. Further, because T is a sufficient statistic, ${\displaystyle {\delta }_1(X)}$ does not depend on θ, hence, becomes a statistic.

This result is known as the Rao–Blackwell theorem.

Risk aversion

The relation between risk aversion and declining marginal utility for scalar outcomes can be stated formally with Jensen's inequality: risk aversion can be stated as preferring a certain outcome ${\displaystyle u(E[x])}$ to a fair gamble with potentially larger but uncertain outcome of ${\displaystyle u(x)}$:

${\displaystyle u(E[x])>E[u(x)]}$.

But this is simply Jensen's inequality for a concave ${\displaystyle u(x)}$: a utility function that exhibits declining marginal utility.

Generalizations

Beyond its classical formulation for real numbers and convex functions, Jensen's inequality has been extended to the realm of operator theory. In this non‐commutative setting the inequality is expressed in terms of operator convex functions—that is, functions defined on an interval I that satisfy

${\displaystyle f(\lambda x+(1-\lambda )y)\le \lambda f(x)+(1-\lambda )f(y)}$

for every pair of self‐adjoint operators x and y (with spectra in I) and every scalar ${\displaystyle \lambda \in [0,1]}$. Hansen and Pedersen established a definitive version of this inequality by considering genuine non‐commutative convex combinations. In particular, if one has an n‑tuple of bounded self‐adjoint operators ${\displaystyle x_1,\dots ,x_n}$ with spectra in I and an n‑tuple of operators ${\displaystyle a_1,\dots ,a_n}$ satisfying

${\displaystyle \sum _{i=1}^n{a}_i^{\ast }{a}_i=I,}$

then the following operator Jensen inequality holds:

${\displaystyle f(\sum _{i=1}^n{a}_i^{\ast }{x}_i{a}_i)\le \sum _{i=1}^n{a}_i^{\ast }f(x_i)a_i.}$

This result shows that the convex transformation “respects” non-commutative convex combinations, thereby extending the classical inequality to operators without the need for additional restrictions on the interval of definition. A closely related extension is given by the Jensen trace inequality. For a continuous convex function f defined on I, if one considers self‐adjoint matrices ${\displaystyle x_1,\dots ,x_n}$ (with spectra in I) and matrices ${\displaystyle a_1,\dots ,a_n}$ satisfying ${\displaystyle \sum _{i=1}^n{a}_i^{\ast }{a}_i=I}$, then one has

${\displaystyle \mathrm{Tr}(f(\sum _{i=1}^n{a}_i^{\ast }{x}_i{a}_i))\le \mathrm{Tr}(\sum _{i=1}^n{a}_i^{\ast }f(x_i)a_i).}$

This inequality naturally extends to C*-algebras equipped with a finite trace and is particularly useful in applications ranging from quantum statistical mechanics to information theory. Furthermore, contractive versions of these operator inequalities are available when one only assumes ${\displaystyle \sum _{i=1}^n{a}_i^t{a}_i\le I}$, provided that additional conditions such as ${\displaystyle f(0)\le 0}$ (when 0 ∈ I) are imposed. Extensions to continuous fields of operators and to settings involving conditional expectations on C-algebras further illustrate the broad applicability of these generalizations.