Conditional jensen inequality
WebDec 24, 2024 · STA 711 Week 5 R L Wolpert Theorem 1 (Jensen’s Inequality) Let ϕ be a convex function on R and let X ∈ L1 be integrable. Then ϕ E[X]≤ E ϕ(X) One proof with a nice geometric feel relies on finding a tangent line to the graph of ϕ at the point µ = E[X].To start, note by convexity that for any a < b < c, ϕ(b) lies below the value at x = b of the … WebNov 17, 2024 · Equality in Conditional Jensen's Inequality. 1. Jensen's inequality and conditional expectation. 4. Does Jensen's inequality still hold in general finite measure …
Conditional jensen inequality
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Webtionals : K!R are given. Moreover, a few applications of the conditional Jensen’s inequality are presented. 2 Preliminary Result. The auxiliary result below is a conditional analogue … 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. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto … See more The classical form of Jensen's inequality involves several numbers and weights. The inequality can be stated quite generally using either the language of measure theory or (equivalently) probability. In the … See more Form involving a probability density function Suppose Ω is a measurable subset of the real line and f(x) is a non-negative function such that $${\displaystyle \int _{-\infty }^{\infty }f(x)\,dx=1.}$$ See more • Jensen's Operator Inequality of Hansen and Pedersen. • "Jensen inequality", Encyclopedia of Mathematics, EMS Press, 2001 [1994] See more Jensen's inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be offered. Before embarking on these … See more • Karamata's inequality for a more general inequality • Popoviciu's inequality • Law of averages • A proof without words of Jensen's inequality See more
WebAgain, conditional Jensen’s inequality follows almost directly fromTheorem 5.5: Corollary 5.7 (conditional Jensen’s inequality). LetAssumption 5.1hold and f: Rd!R be a convex function. Then E[f(X)jF] f(E[XjF]) P-almost surely. Proof. This follows directly fromTheorem 5.5andLemma 2.1. Acknowledgments WebMay 16, 2024 · Relative entropy is a well-known asymmetric and unbounded divergence measure [], whereas the Jensen-Shannon divergence [19,20] (a.k.a. the capacitory discrimination []) is a bounded symmetrization of relative entropy, which does not require the pair of probability measures to have matching supports.It has the pleasing property that …
Webin Section 14, but so far we’ve proved them only for p = q = 2 (for H¨older’s inequality) and for p = 1 or p = 2 (for Minkowski’s inequality). In this section we provide proofs for general p. We also discuss Jensen’s inequality, which is especially important in Probability theory. These proofs are non-examinable. WebJensen’s Inequality: Let C Rdbe convex and suppose that X2C. Provided that all expectations are well-defined, the following hold. (1)The expectation EX2C (2)If f: C!R is convex then f(EX) Ef(X). If fis strictly convex and Xis not constant then the inequality is strict. (3)If f: C!R is concave then f(EX) Ef(X). If fis strictly concave and Xis
WebJensen's inequality is a powerful mathematical tool and one of the workhorses in statistical learning. Its applications therein include the EM ... maximum conditional likelihood, large margin discriminative models and conditional Bayesian inference. Convergence, efficiency and prediction results are shown. 1
WebMar 24, 2024 · Jensen-like inequalities are introduced, as well as a generalisation of a recent improvement to Jensen's inequality. Some of their applications are proposed: extensions of Lyapunov's inequality and inferential problems. ... [15] Pelessoni R., Vicig P., 2-coherent and 2-convex conditional lower previsions, Int. J. Approx. Reason. 77 ... haynes michaelWebNov 16, 2024 · Sorted by: 2. A generalized Jensen inequality for a conditional expectation of a random function (with values in an ordered Banach space) of a random vector in another Banach space is given here, with some review of the prehistory. The proof does not seem trivial; it takes over three pages and contains references to previous results. bottles on the wall dragon age inquisitionWebSeveral properties of entropy follow from Jensen's inequality. We give a proof for the case of finite sums: Theorem (Jensen's inequality) Suppose f is continuous strictly concave … bottles oothttp://www.ece.tufts.edu/ee/194NIT/lect01.pdf haynes middle school nashvilleWebApr 17, 2024 · I have some doubts as I read over Durrett's proof on Jensen's inequality for conditional expectation. The statement is that if $\varphi$ is convex and $E X , … haynes middle school metairieWebJensen's inequality for conditional expectations (PDF) Jensen's inequality for conditional expectations Frank Hansen - Academia.edu Academia.edu no longer supports Internet Explorer. haynes middle school nashville scheduleWebApr 10, 2024 · Graph Convex Hull Bounds as generalized Jensen Inequalities. Jensen's inequality is ubiquitous in measure and probability theory, statistics, machine learning, information theory and many other areas of mathematics and data science. It states that, for any convex function on a convex domain and any random variable taking values in , . haynes middle school nashville tn