Jensen inequality
E87727
Jensen's inequality is a fundamental result in convex analysis and probability theory that relates the value of a convex (or concave) function of an expectation to the expectation of the function, providing bounds that underlie many other inequalities and convergence results.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Jensen inequality canonical | 5 |
| Jensen's inequality | 5 |
| Jensen functional equation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T736584 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Jensen inequality Context triple: [Minkowski inequality, relatedTo, Jensen inequality]
-
A.
Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
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B.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
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C.
Rényi divergence
Rényi divergence is a family of information-theoretic measures that generalize Kullback–Leibler divergence to quantify the dissimilarity between probability distributions, parameterized by an order α.
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D.
Karush–Kuhn–Tucker conditions
The Karush–Kuhn–Tucker conditions are fundamental optimality criteria in nonlinear programming that generalize Lagrange multipliers to handle inequality constraints.
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E.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Jensen inequality Target entity description: Jensen's inequality is a fundamental result in convex analysis and probability theory that relates the value of a convex (or concave) function of an expectation to the expectation of the function, providing bounds that underlie many other inequalities and convergence results.
-
A.
Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
-
B.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
-
C.
Rényi divergence
Rényi divergence is a family of information-theoretic measures that generalize Kullback–Leibler divergence to quantify the dissimilarity between probability distributions, parameterized by an order α.
-
D.
Karush–Kuhn–Tucker conditions
The Karush–Kuhn–Tucker conditions are fundamental optimality criteria in nonlinear programming that generalize Lagrange multipliers to handle inequality constraints.
-
E.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
- F. None of above. chosen
Statements (56)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical inequality
ⓘ
result in convex analysis ⓘ result in probability theory ⓘ |
| appliesTo |
concave functions
ⓘ
convex functions ⓘ |
| coreStatement |
For a concave function φ and random variable X, φ(E[X]) ≥ E[φ(X)]
ⓘ
For a convex function φ and random variable X, φ(E[X]) ≤ E[φ(X)] ⓘ |
| equalityCondition |
convex function is affine on the support of the random variable
ⓘ
random variable is almost surely constant ⓘ |
| field |
convex analysis
ⓘ
measure theory ⓘ probability theory ⓘ real analysis ⓘ |
| generalizationOf |
Cauchy–Schwarz inequality in some formulations
ⓘ
inequality between arithmetic and geometric means ⓘ inequality between arithmetic and harmonic means ⓘ |
| hasVariant |
conditional Jensen's inequality
ⓘ
matrix Jensen inequality ⓘ operator Jensen inequality ⓘ |
| holdsFor |
continuous distributions
ⓘ
discrete distributions ⓘ finite sums ⓘ integrals ⓘ probability measures ⓘ |
| implies |
E[|X|^p] ≥ |E[X]|^p for p ≥ 1
ⓘ
log E[X] ≥ E[log X] for positive X and concave log ⓘ |
| namedAfter | Johan Jensen ⓘ |
| relatedTo |
Gibbs' inequality
ⓘ
linked to:
Kullback–Leibler divergence
Karamata's inequality ⓘ Young's inequality ⓘ convex combination ⓘ epigraph of a convex function ⓘ majorization theory ⓘ supporting hyperplane ⓘ |
| relates |
expectation of a function
ⓘ
expectation of a random variable ⓘ function of an expectation ⓘ |
| requires |
convexity of the function on the range of the random variable
ⓘ
integrable random variable ⓘ |
| timePeriod | early 20th century ⓘ |
| usedFor |
Hölder-type inequalities
ⓘ
Jensen–Shannon divergence properties ⓘ Kullback–Leibler divergence inequalities ⓘ
linked to:
Kullback–Leibler divergence
Minkowski inequality proofs ⓘ bounding expectations ⓘ bounding moments of random variables ⓘ convex optimization analysis ⓘ deriving other inequalities ⓘ entropy bounds ⓘ evidence lower bound (ELBO) derivation ⓘ information theory inequalities ⓘ machine learning generalization bounds ⓘ proving convergence results ⓘ proving law of large numbers variants ⓘ risk measures in finance ⓘ variational inference ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Jensen inequality Description of subject: Jensen's inequality is a fundamental result in convex analysis and probability theory that relates the value of a convex (or concave) function of an expectation to the expectation of the function, providing bounds that underlie many other inequalities and convergence results.
Referenced by (11)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Jensen inequality
linked to: Jensen inequality
linked to: Jensen inequality
linked to: Jensen inequality
linked to: Jensen inequality
linked to: Jensen inequality