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Jensen's Inequality

For a convex function, the function of a mean is no greater than the mean of the function's values, under the required domain and expectation conditions.

Version
v1 · 2026-10-03 · History
Domain-specific #
13353
Aliases
Jensen Inequality

Core Idea

For a convex function \(f\) and a random input \(X\) with suitable domain and expectation conditions, Jensen's inequality states \(f(E[X])\leq E[f(X)]\). A concave function reverses the inequality. It compares transforming a mean with averaging transformed values under the same law. The gap is nonnegative for convex \(f\), but neither a strict gap nor an exact curvature–variance formula is universal. Equality can hold with nonconstant \(X\) when \(f\) is affine on the relevant supported mixtures; strict convexity at the mean supports the stronger constant-input equality conclusion.[ref-f9c77b413d66][ref-6e5fb9e6c3d6]

Scope of Application

MIT applies the concave form to an increasing expected-utility function \(u\) for wealth lotteries: \(E[u(X)]\leq u(E[X])\). This is a utility comparison, not a monetary risk premium; the latter needs a certainty equivalent \(u^{-1}(E[u(X)])\). Cover and Thomas apply log concavity to density ratios to prove relative entropy nonnegative. Their finite Jensen argument works on the positive-support ratio; if the reference density vanishes on positive target mass, divergence is infinite by convention.[ref-a8fa868a3274][ref-ef385416fd6e]

Clarity

State the probability law or normalized weights, the convexity/concavity domain, and whether all expressions are defined. A curved response alone does not determine the direction. A zero gap does not prove constant input under mere convexity, and a utility-valued gap cannot be relabeled as wealth.[ref-6e5fb9e6c3d6][ref-a8fa868a3274]

Manages Complexity

Convexity turns a potentially difficult expectation of nonlinear values into a directional bound without computing the full integral. The tradeoff is coarseness: it gives an order relation, not a general numeric gap. Distinct applications inherit this comparison while retaining their own support, unit and interpretation conditions.[ref-f9c77b413d66][ref-ef385416fd6e]

Abstract Reasoning

Verify the law or weights, finite relevant expectations, a convex domain containing the inputs and mean, and convexity on that domain. Only then compare \(f(E[X])\) with \(E[f(X)]\). For strict equality claims, add strict-convexity hypotheses; for KL claims, check where the reference density is zero before using finite log ratios.[ref-6e5fb9e6c3d6][ref-ef385416fd6e]

Knowledge Transfer

The same mean/transform ordering transfers literally from concave wealth utility to log-ratio information theory, but neither economic risk preferences nor density support conventions transfer automatically. Live prime Convexity supplies the broader chord-dominance skeleton. A direct DAG edge is deferred because its current live ancestry incorrectly makes Optimization a prerequisite to this theorem.[ref-f9c77b413d66][ref-a8fa868a3274][^ref-ef385416fd6e]

[^ref-f9c77b413d66]: Stephen Boyd and Lieven Vandenberghe, Convex Optimization I, Stanford EE364a lecture slides, Convex Functions 3–12, PDF p. 49. [^ref-6e5fb9e6c3d6]: John Duchi, Exercises for Theory of Statistics, Stanford Stats300b (Winter 2021), §2 Questions 2.1(e)–2.2, PDF p. 7. [^ref-a8fa868a3274]: Alexander Wolitzky, Lecture 9: Attitudes toward Risk, MIT 14.121 (Fall 2015), PDF pp. 3–6. [^ref-ef385416fd6e]: Thomas M. Cover and Joy A. Thomas, “Determinant Inequalities via Information Theory”, SIAM Journal on Matrix Analysis and Applications 9(3) (1988), §2 Lemma 1, PDF pp. 1–2.

Neighborhood in Abstraction Space

Jensen's Inequality sits in a sparse region of the domain-specific corpus (70th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Foundations of Probability & Inference (29 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08