en.unionpedia.org

Jensen's inequality, the Glossary

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.^[1]

53 relations: Acta Mathematica, AM–GM inequality, Concave function, Conditional expectation, Continuous function, Convex combination, Convex function, Corollary, Counting measure, Degenerate distribution, Dense set, Dirac delta function, Dual space, Empty set, Estimator, Expected value, Function (mathematics), Gibbs' inequality, Graph of a function, Integral, Σ-algebra, Johan Jensen (mathematician), Karamata's inequality, Kullback–Leibler divergence, Law of averages, Logarithm, Marginal utility, Mathematical induction, Mathematical proof, Mathematics, Measure (mathematics), Moment (mathematics), Number line, Otto Hölder, Popoviciu's inequality, Probability density function, Probability distribution, Probability space, Probability theory, Random variable, Rao–Blackwell theorem, Risk aversion, Sample space, Secant line, Subderivative, Sufficient statistic, The American Mathematical Monthly, The American Statistician, Topological vector space, Tristan Needham, ... Expand index (3 more) »
Statistical inequalities
Theorems involving convexity

Acta Mathematica

Acta Mathematica is a peer-reviewed open-access scientific journal covering research in all fields of mathematics.

See Jensen's inequality and Acta Mathematica

In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which case they are both that number). Jensen's inequality and AM–GM inequality are Inequalities.

See Jensen's inequality and AM–GM inequality

Concave function

In mathematics, a concave function is one for which the value at any convex combination of elements in the domain is greater than or equal to the convex combination of the values at the endpoints. Jensen's inequality and concave function are convex analysis.

See Jensen's inequality and Concave function

Conditional expectation

In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution.

See Jensen's inequality and Conditional expectation

Continuous function

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function.

See Jensen's inequality and Continuous function

Convex combination

In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1.

See Jensen's inequality and Convex combination

Convex function

In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. Jensen's inequality and convex function are convex analysis.

See Jensen's inequality and Convex function

Corollary

In mathematics and logic, a corollary is a theorem of less importance which can be readily deduced from a previous, more notable statement.

See Jensen's inequality and Corollary

Counting measure

In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity \infty if the subset is infinite.

See Jensen's inequality and Counting measure

Degenerate distribution

In mathematics, a degenerate distribution (sometimes also Dirac distribution) is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with support only at a single point.

See Jensen's inequality and Degenerate distribution

Dense set

In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).

See Jensen's inequality and Dense set

Dirac delta function

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.

See Jensen's inequality and Dirac delta function

Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

See Jensen's inequality and Dual space

Empty set

In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

See Jensen's inequality and Empty set

Estimator

In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished.

See Jensen's inequality and Estimator

Expected value

In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average.

See Jensen's inequality and Expected value

Function (mathematics)

In mathematics, a function from a set to a set assigns to each element of exactly one element of.

See Jensen's inequality and Function (mathematics)

Gibbs' inequality

Josiah Willard Gibbs In information theory, Gibbs' inequality is a statement about the information entropy of a discrete probability distribution. Jensen's inequality and Gibbs' inequality are Probabilistic inequalities.

See Jensen's inequality and Gibbs' inequality

Graph of a function

In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x).

See Jensen's inequality and Graph of a function

Integral

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations.

See Jensen's inequality and Integral

Σ-algebra

In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections.

See Jensen's inequality and Σ-algebra

Johan Jensen (mathematician)

Johan Ludwig William Valdemar Jensen, mostly known as Johan Jensen (8 May 1859 – 5 March 1925), was a Danish mathematician and engineer.

See Jensen's inequality and Johan Jensen (mathematician)

Karamata's inequality

In mathematics, Karamata's inequality, named after Jovan Karamata, also known as the majorization inequality, is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line. Jensen's inequality and Karamata's inequality are convex analysis and Inequalities.

See Jensen's inequality and Karamata's inequality

Kullback–Leibler divergence

In mathematical statistics, the Kullback–Leibler (KL) divergence (also called relative entropy and I-divergence), denoted D_\text(P \parallel Q), is a type of statistical distance: a measure of how one probability distribution is different from a second, reference probability distribution.

See Jensen's inequality and Kullback–Leibler divergence

Law of averages

The law of averages is the commonly held belief that a particular outcome or event will, over certain periods of time, occur at a frequency that is similar to its probability.

See Jensen's inequality and Law of averages

Logarithm

In mathematics, the logarithm is the inverse function to exponentiation.

See Jensen's inequality and Logarithm

Marginal utility

In economics, marginal utility describes the change in utility (pleasure or satisfaction resulting from the consumption) of one unit of a good or service.

See Jensen's inequality and Marginal utility

Mathematical induction

Mathematical induction is a method for proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots  all hold.

See Jensen's inequality and Mathematical induction

Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.

See Jensen's inequality and Mathematical proof

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Jensen's inequality and Mathematics

Measure (mathematics)

In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events.

See Jensen's inequality and Measure (mathematics)

Moment (mathematics)

In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph.

See Jensen's inequality and Moment (mathematics)

Number line

In elementary mathematics, a number line is a picture of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely.

See Jensen's inequality and Number line

Otto Hölder

Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart.

See Jensen's inequality and Otto Hölder

Popoviciu's inequality

In convex analysis, Popoviciu's inequality is an inequality about convex functions. Jensen's inequality and Popoviciu's inequality are convex analysis and Inequalities.

See Jensen's inequality and Popoviciu's inequality

Probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would be equal to that sample.

See Jensen's inequality and Probability density function

Probability distribution

In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of possible outcomes for an experiment.

See Jensen's inequality and Probability distribution

Probability space

In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment".

See Jensen's inequality and Probability space

Probability theory

Probability theory or probability calculus is the branch of mathematics concerned with probability.

See Jensen's inequality and Probability theory

Random variable

A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events.

See Jensen's inequality and Random variable

Rao–Blackwell theorem

In statistics, the Rao–Blackwell theorem, sometimes referred to as the Rao–Blackwell–Kolmogorov theorem, is a result that characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria.

See Jensen's inequality and Rao–Blackwell theorem

Risk aversion

In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more certain outcome.

See Jensen's inequality and Risk aversion

References

[1] https://en.wikipedia.org/wiki/Jensen's_inequality

Also known as Jensen Inequality.

, Utility, Variational Bayesian methods, Weak topology.

Jensen's inequality, the Glossary

Table of Contents

See also

Statistical inequalities

Theorems involving convexity

References