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. 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 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, while the graph of the function is the convex function of the weighted means,
- 1 Statements
- 2 Proofs
- 3 Applications and special cases
- 4 See also
- 5 Notes
- 6 References
- 7 External links
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 x1, x2, ..., xn in its domain, and positive weights ai, Jensen's inequality can be stated as:
and the inequality is reversed if is concave, which is
As a particular case, if the weights ai are all equal, then (1) and (2) become
For instance, the function log(x) is concave, so substituting in the previous formula (4) establishes the (logarithm of) the familiar arithmetic mean-geometric mean inequality:
The variable x may, if required, be a function of another variable (or set of variables) t, so that xi = g(ti). All of this carries directly over to the general continuous case: the weights ai are replaced by a non-negative integrable function f(x), such as a probability distribution, and the summations are replaced by integrals.
Measure-theoretic and probabilistic form
In real analysis, we may require an estimate on
where are real numbers, and is a non-negative real-valued function that is Lebesgue-integrable. In this case, the Lebesgue measure of need not be unity. However, by integration by substitution, the interval can be rescaled so that it has measure unity. Then Jensen's inequality can be applied to get
The same result can be equivalently stated in a probability theory setting, by a simple change of notation. Let be a probability space, X an integrable real-valued random variable and a convex function. Then:
General inequality in a probabilistic setting
More generally, let T be a real topological vector space, and X a T-valued integrable random variable. In this general setting, integrable means that there exists an element in T, such that for any element z in the dual space of T: , and . Then, for any measurable convex function φ and any sub-σ-algebra of :
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 and its image in the graph. Noticing that for convex mappings the corresponding distribution of Y values is increasingly "stretched out" 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 . Consequently, in this picture the expectation of Y will always shift upwards with respect to the position of , and this "proves" the inequality, i.e.
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 implies
This can be easily generalized: if λ1, λ2, ..., λn are nonnegative real numbers such that λ1 + ... + λn = 1, then
for any x1, ..., xn. This finite form of the Jensen's inequality can be proved by induction: by convexity hypotheses, the statement is true for n = 2. Suppose it is true also for some n, one needs to prove it for n + 1. At least one of the λi is strictly positive, say λ1; therefore by convexity inequality:
Since , one can apply the induction hypotheses to the last term in the previous formula to obtain the result, namely the finite form of the Jensen's inequality.
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:
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 g be a real-valued μ-integrable function on a probability space Ω, and let φ be a convex function on the real numbers. Since φ is convex, at each real number x we have a nonempty set of subderivatives, which may be thought of as lines touching the graph of φ at x, but which are at or below the graph of φ at all points.
Now, if we define
because of the existence of subderivatives for convex functions, we may choose an a and b such that
for all real x and
But then we have that
for all x. Since we have a probability measure, the integral is monotone with μ(Ω)=1 so that
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 is convex, for any , the quantity
is decreasing as θ approaches 0+. In particular, the subdifferential of φ evaluated at x in the direction y is well-defined by
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
In particular, for an arbitrary sub-σ-algebra we can evaluate the last inequality when to obtain
Now, if we take the expectation conditioned to on both sides of the previous expression, we get the result since:
by the linearity of the subdifferential in the y variable, and the following well-known property of the conditional expectation:
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
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 φ is convex over the range of g, then
If g(x) = x, then this form of the inequality reduces to a commonly used special case:
Alternative finite form
If is some finite set , and if is a counting measure on , then the general form reduces to a statement about sums:
There is also an infinite discrete form.
Jensen's inequality is of particular importance in statistical physics when the convex function is an exponential, giving:
The proof in this case is very simple (cf. Chandler, Sec. 5.5). The desired inequality follows directly, by writing
and then applying the inequality
to the final exponential.
If p(x) is the true probability distribution for x, and q(x) is another distribution, then applying Jensen's inequality for the random variable Y(x) = q(x)/p(x) and the function (y) = −log(y) gives
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.
Since -log(x) is a strictly convex function for x>0, it follows that equality holds when p(x) equals q(x) almost everywhere.
If L is a convex function, then from Jensen's inequality we get
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
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.
This result is known as the Rao–Blackwell theorem.
|This article needs additional citations for verification. (October 2011)|
- Jensen, J. L. W. V. (1906). "Sur les fonctions convexes et les inégalités entre les valeurs moyennes". Acta Mathematica 30 (1): 175–193. doi:10.1007/BF02418571.
- David Chandler (1987). Introduction to Modern Statistical Mechanics. Oxford. ISBN 0-19-504277-8.
- Tristan Needham (1993) "A Visual Explanation of Jensen's Inequality", American Mathematical Monthly 100(8):768–71.
- Walter Rudin (1987). Real and Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1.
- Jensen's Operator Inequality of Hansen and Pedersen.
- Hazewinkel, Michiel, ed. (2001), "Jensen inequality", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W., "Jensen's inequality", MathWorld.
- Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format.