How do you use Jensen inequality?
How do you use Jensen inequality?
To use Jensen’s inequality, we need to determine if a function g is convex. A useful method is the second derivative. A twice-differentiable function g:I→R is convex if and only if g″(x)≥0 for all x∈I.
How do you prove Jensen’s inequality?
Jensen’s inequality states that this line is everywhere at least as large as f(x). pf(x1) + (1 − p)f(x2) ≥ f(px1 + (1 − p)x2). If f is (doubly) differentiable then f is convex if and only if d2f/dx2 ≥ 0.
Is expected value convex?
In probability theory, a convex function applied to the expected value of a random variable is always bounded above by the expected value of the convex function of the random variable.
How do you show a function is convex?
A function f : Rn → R is convex if and only if the function g : R → R given by g(t) = f(x + ty) is convex (as a univariate function) for all x in domain of f and all y ∈ Rn. (The domain of g here is all t for which x + ty is in the domain of f.) Proof: This is straightforward from the definition.
Why is Jensen’s inequality important?
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 mathematical proofs. It is also used to make claims about a function where little is known or needs to be known about the distribution.
Is log concave or convex?
Logarithm is Strictly Concave – ProofWiki.
Is sqrt convex?
The sqrt function is concave and increasing, which means it can only take a concave argument. In fact, sqrt(1 + square(x)) is a convex function of x , but the DCP rules are not able to verify convexity.
Does expectation preserve concavity?
In recent years Karlin [ 4] has shown that the concavity of a function is preserved under the expectation transformation with respect to a class of totally positive distributions of order S. This note extends a result of Karlin to a bigger class of distributions.
How do you know if an inequality is convex?
1. If you know calculus, take the second derivative. It is a well-known fact that if the second derivative f (x) is ≥ 0 for all x in an interval I, then f is convex on I. On the other hand, if f(x) ≤ 0 for all x ∈ I, then f is concave on I.
How do you know if a problem is convex?
Convex Functions Algebraically, f is convex if, for any x and y, and any t between 0 and 1, f( tx + (1-t)y ) <= t f(x) + (1-t) f(y). A function is concave if -f is convex — i.e. if the chord from x to y lies on or below the graph of f.
What is Jensen’s integral inequality for a convex function?
Jensen’s integral inequality for a convex function f is: ∫ D λ ( t) d t = 1. Equality holds if and only if either x ( t) = const on D or if f is linear on x ( D) . If f is a concave function, the inequality signs in (1) and (2) must be reversed.
What is the significance of the Jensen inequality?
Jensen’s inequality generalizes the statement that a secant line of a convex function lies above the graph. 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 proven by Jensen in 1906.
Are there different forms of the inequality?
Given its generality, the inequality appears in many forms depending on the context, some of which are presented below.