The jensen inequality
WebJun 5, 2024 · Equality holds if and only if $ x _ {1} = \dots = x _ {n} $ or if $ f $ is linear. Jensen's integral inequality for a convex function $ f $ is: $$ \tag{2 } f \left ( \int\limits _ { … WebFeb 1, 2016 · This inequality can be traced back to Jensen’s original papers [ 1, 2] and is one of the most fundamental mathematical inequalities. One reason for that is that in fact a great number of classical inequalities can be derived from ( 1.1 ), see e.g. [ 3] and the references given therein.
The jensen inequality
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WebDec 10, 2024 · The Jensen inequality plays an important role in obtaining inequalities for divergences between probability distributions, and there are many papers dealing with inequalities for divergences and entropies (see, e.g., [7, 9, 12]). In this section, we give some applications of our results, and we first introduce the basic notions. WebDec 10, 2024 · The Jensen inequality plays an important role in obtaining inequalitie s for divergences between probability distributions, and there are many papers dealing with inequalitie s for divergences and ...
WebExamples of Jensen inequalities. The most familiar example of a Jensen inequality occurs when the weights are all equal to 1/ N and the convex function is f ( x) = x2 . In this case the Jensen inequality gives the familiar result that the mean square exceeds the square of the mean: In the other applications we will consider, the population ... Webas generalized Jensen Inequalities Ilja Klebanov1 April 12, 2024 Abstract. Jensen’s inequality is ubiquitous in measure and probability theory, statistics, machine learning, information theory and many other areas of mathematics and data sci-ence. It states that, for any convex function f: K!R on a convex domain K Rd and
WebFeb 24, 2015 · It was established by the Danish mathematician JLWV Jensen in [ 1 ], and it is important in convex analysis [ 2, 3 ], since it can be used to generalize the triangle inequality. In addition, the Jensen inequality is an important tool in L^ {p} -spaces and in connection to modular and Orlicz spaces; see, e.g., [ 4, 5 ]. WebWhen Jensen's inequality is equality. If X is a random variable and g is a convex function, then E ( g ( X)) ≥ g ( E ( X)). Just out of curiosity, when do we have equality? If and only if g …
WebJensen's inequality by Marco Taboga, PhD Jensens's inequality is a probabilistic inequality that concerns the expected value of convex and concave transformations of a random …
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 … league of kansas municipalities budget tipsWebDec 1, 2024 · My goal is to show this inequality: 1 + ( E ( X)) 2 ≤ E ( 1 + X 2) x² is a convex function, so with the Jensen inequality I get that: 1 + ( E ( X)) 2 ≤ 1 + E ( X 2) = E ( 1 + X 2) But when I use the Jensen inequality a second time, for the concave function, I get that: 1 + ( E ( X)) 2 ≤ 1 + E ( X 2) = E ( 1 + X 2) ≥ E ( 1 + X 2) league of ireland previewWebNov 12, 2024 · The Jensen inequality is a widely used tool in a multitude of fields, such as for example information theory and machine learning. It can be also used to derive other standard inequalities such as the inequality of arithmetic and geometric means or the Hölder inequality. In a probabilistic setting, the Jensen inequality describes the … league of ireland scheduleWebJensen Inequality Theorem 1. Let fbe an integrable function de ned on [a;b] and let ˚be a continuous (this is not needed) convex function de ned at least on the set [m;M] where … league of jobWebDec 1, 2024 · My goal is to show this inequality: 1 + ( E ( X)) 2 ≤ E ( 1 + X 2) x² is a convex function, so with the Jensen inequality I get that: 1 + ( E ( X)) 2 ≤ 1 + E ( X 2) = E ( 1 + X 2) … league of kingdoms - lokaWebJensen's Inequality appears multiple times in any rigorous machine learning textbook. It's essential for the key principles and foundational algorithms that make this field so productive. In... league of joy how to drawWeb1 Answer. Yes Jensen inequality holds for multiple variable. We can find a general formulation in the mesure theoretic article in Wikipedia. "Let (Ω, A, μ) be a measure space, such that μ (Ω) = 1. If g is a real-valued function that is μ-integrable, and if g is a convex function on the real line. This generalizable for convex function on ... league of ireland underage fixtures