Jensen inequality pdf

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August undefined, 2024

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 … Webn Jensen’s inequality states: f(w 1x 1 +w 2x 2 +:::w nx n) w 1f(x 1)+w 2f(x 2)+:::+w nf(x n) Proof We proceed by induction on n, the number of weights. If n= 1 then equality holds …

1.2. Distribution, expectation and inequalities. - Hong Kong …

WebUsing Jensen’s inequality, we can derive a bound, which is extremely useful in the EM algorithm. It is also important to know when the equality holds in Jensen’s inequality: f(E[x]) = E[f(x)], i x is a constant. 5 EM algorithm Given training data set fx(1);:::;x(m)g. We want to nd parameters to t a model p(x;z; ), where z is latent variables. WebThe following is a useful inequality for dealing with the entropy function and its derivatives: Lemma 5 (Jensen’s Inequality). If f is a convex function on (a;b) and Xis a random variable taking values in (a;b), then f(E[X]) E[f(X)] Proof We prove the case when Xtakes on nitely many values. The general case follows by continuity arguments. how to glue a jigsaw puzzle to hang on wall

Olympiad Corner Jensen’s Inequality - Hong Kong University of …

WebJensen’s Inequality Theorem For any concave function f, E[f(X)] f(E[X]) Proof. Suppose f is di erentiable. The function f is concave if, for any x and y, WebJul 31, 2024 · Jensen’s Inequality is a useful tool in mathematics, specifically in applied fields such as probability and statistics. For example, it is often used as a tool in … WebAlthough many of the inequalities here can be stated in much more general terms after the basics about measure and integration are developed, the mechanisms for these … john sowa chicago

Refinements of Jensen’s inequality by uniformly convex functions

Jensen’s Inequality - Electrical Engineering and …

WebThe Jensen inequality is one of the most important inequalities in theory of inequalities, and numerous results are devoted to this inequality. This inequality has many applications in several fields. WebJENSEN INEQUALITY BASICS This is a revision of material that appeared earlier in SEP 37 and reprinted in PVI. Let f be a function with a positive second derivative. Such a function … how to glue a golf head on a shaftWebExpectation Inequalities JENSEN’S INEQUALITY Jensen’s Inequality gives a lower bound on expectations of convex functions. Recall that a function g(x) is convex if, for 0 < < 1, g( x+(1 )y) g(x)+(1 )g(y) for all x and y. Alternatively, if the derivatives are well deﬁned, function g(x) is convex if d2 dt2 how to glue a jigsaw puzzle together

"WebTutorial 8: Jensen inequality 6 Deﬁnition 67 Let (Ω,T) be a topological space. We say that (Ω,T) is a Hausdorﬀ topological space, if and only if for all x,y ∈ Ω with x = y, there exists … " - Jensen inequality pdf

Jensen inequality pdf

A Note on Some New Refinements of Jensen

Webinequalities than the standard exposition Olympiad Inequalities, by Thomas Mildorf. I was motivated to write it by feeling guilty for getting free 7’s on ... Theorem 2.1 (Jensen’s … WebThe Jensen inequality [9] is of great interest in differential and difference equations, and other areas of mathematics. The original Jensen inequality is as follows: Ifg 2 C([a;b];(c;d)) andf 2 C((c;d);R) are convex, then f ˆR b a g(s)ds b¡a Rb a f(g(s))ds b¡a :

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WebJul 6, 2010 · In this chapter, we shall establish Jensen's inequality, the most fundamental of these inequalities, in various forms. A subset C of a real or complex vector space E is … WebArithmetic and geometric means satisfy a famous inequality, namely that the geometric mean is always less than or equal to the arithmetic mean. This turns out to be a simple application of Jensen’s inequality: Theorem 5 AM{GM Inequality Let x 1;:::;x n>0, and let 1;:::; n2[0;1] so that 1 + + n= 1. Then x 1 1 x n n 1x 1 + + nx n:

WebSep 27, 2000 · Jensen’s Inequality becomes equality only when n = 1 or function U is affine-linear over at least the convex hull of the given arguments xj; can you see why? (It takes a while.) Jensen’s Inequality has many applications. An important one is the Inequality among the Arithmetic, Geometric and Harmonic Means: ... http://sepwww.stanford.edu/data/media/public/sep/jon/jensen.pdf

WebOct 31, 2024 · (PDF) GENERALIZATION AND REFINEMENTS OF JENSEN INEQUALITY Home Refining GENERALIZATION AND REFINEMENTS OF JENSEN INEQUALITY Authors: Faiza Rubab Hira Nabi Asif R. Khan University of... Webt. Jensen’s inequality says that f( 1x 1 + 2x 2 + + nx n) 1f(x 1) + 2f(x 2) + + nf(x n): When x 1;x 2;:::;x n are not all equal, because fis strictly convex, we get a >in this inequality. That’s …

WebIn this note, we obtain two new refinements of Jensen's inequality for convex functions.

WebJensen’s Inequality Konstantinos G. Derpanis [email protected] Version 1.0 March 12, 2005 In this note the concept of convexity and Jensen’s Inequality are reviewed. Jensen’s … john sowers obituaryWeb(1) the Jensen inequality: Suppose ψ(·) is a convexfunction and Xand ψ(X) haveﬁnite expectation. Then ψ(E(X)) ≤ E(ψ(X)). Proof. Convexity implies for every a, there exists a … how to glue and nail hardwood flooringhttp://www.cse.yorku.ca/~kosta/CompVis_Notes/jensen.pdf how to glue aluminum to woodWebThe classical Jensen inequality is a famous tool to construct new results in the theory of inequalities. It has numerous applications in abstract and applied sciences. In this … john sowa home pageWebIn mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex functionof an integralto the integral of the convex function. It … how to glue aluminum to glassWeb3.1 Jensen’s Inequality Here we shall state and prove a generalized, measure theoretic proof for Jensen’s inequality. In general, in probability theory, a more speciﬁc form of Jensen’s inequality is famous. But before that we shall ﬁrst deﬁne a con-vex function. Deﬁnition A function (x) is deﬁned to be convex in interval (a,b ... how to glue a loose floor tileWebity known as Jensen’sinequality (we will see this very important inequality later in our derivation of the EM algorithm). Jensen’s inequality states the following: if f : R→ Ris a convex function, meaning that f is bowl-shaped, then f(E[Z]) ≤ E[f(Z)]. The simplest way to remember this inequality is to think of f(t) = t2, and johnsoyster.com