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 |
Kullback–Leibler divergence
ⓘ
surface form:
Gibbs' inequality
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 ⓘ
surface form:
Kullback–Leibler divergence inequalities
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
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
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.
this entity surface form:
Jensen functional equation
this entity surface form:
Jensen's inequality
this entity surface form:
Jensen's inequality
this entity surface form:
Jensen's inequality
this entity surface form:
Jensen's inequality
this entity surface form:
Jensen's inequality