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.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| Jensen's inequality | 0 |
| Jensen functional equation | 1 |
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 ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Jensen functional equation