WebJul 17, 2024 · Here we have twice used Jensen's inequality for the concave function x ↦ x q − 1. Putting things together. ( E [ X 1]) q = ( a) ( E [ 1 n ∑ i = 1 n X i]) q ≤ ( b) E [ ( 1 n ∑ … 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 …

probability - Equality in Conditional Jensen

WebWell, this proves that if f is a strictly convex function then. E [ f ( X)] = f ( E [ X]) X = E [ X] with prob 1. It uses the standard (nonstrict) Jensen's inequality along the way (applying … WebJensen’s Inequality Jensen’s inequality applies to convex functions. Intuitively a function is convex if it is “upward bending”. f(x) = x2 is a convex function. To make this definition precise consider two real numbers x 1 and x 2. f is convex if the line between f(x 1) and f(x 2) stays above the function f. To make this even brsaoapp24/kpmg_rh/cermain.aspx#

Lecture 1: Entropy and mutual information - Tufts University

WebJensen's inequality is an inequality involving convexity of a function. We first make the following definitions: A function is convex on an interval \(I\) if the segment between any … WebThe inequality is introduced due to the application of Jensen’s inequality and the concavity of log. 3. Divergence is a convex function on the domain of probability distributions. Formally, Lemma 1 (Convexity of divergence). Let p 1;q 1 and p 2;q 2 be probability distributions over a random variable Xand 8 2(0;1) de ne p = p 1 + (1 )p 2 q = q ... WebWe will actually apply generalised Jensen’s inequality with conditional expectations, so we need the following theorem. Theorem A.2 (Generalised Conditional Jensen’s Inequality). Suppose Tis a real Hausdorff locally convex (possibly infinite-dimensional) linear topological space, and let Cbe a closed convex subset of T. Suppose brs and ceramic tiles