Questions tagged [jensen-inequality]
For questions about proving and using Jensen's inequality for convex functions. To be used necessarily with the [inequality] tag.
465
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Relationship between the expectation of the squared L2 norm of sum of elements and the sum of expectations of squared L2 norm of elements.
I am quite confused about the relationship between the following terms,
$E||\sum_{i=1}^n f(x_i) x_i||^2$ and $\sum_{i=1}^n E||f(x_i) x_i||^2$ where $x_i \in \mathbb{R}^p$ and $f(\cdot): \mathbb{R}^p \...
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18
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Ratio of two Jensen's Inequality
I have these pair of numbers
$ (a, b) = (\frac{4}{9}, \frac{1}{9}) $ and $(c, d) = (\frac{1}{2}, \frac{1}{6}) $. (Number mean nothing, just for illustration and simplification)
Note that - (a, b) are ...
- 177
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58
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The inequality of convex function including power function
I want to show the following.
$$ \phi((\lambda g_1 + (1-\lambda)g_2)^k) - (\lambda g_1 + (1-\lambda) g_2)^k \nabla \phi(f^k) \le \lambda \phi(g_1^k) + (1-\lambda)\phi(g_2^k) - (\lambda g_1^k + (1-\...
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1
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39
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Does the Jensen's inequality hold under expectation? [closed]
In other words, does
$$
E\big[E[f(X)]\big] \ge E\big[f(E[X])\big]
$$
holds for a convex function $f(x)$?
If so why? or why not?
1
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1
answer
41
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Is this Jensen's Inequality? (Symmetrization Lemma)
This question concerns most proofs I've seen on the so called Symmetrization Lemma.
Let $\mathcal{F}$ be a class of measurable functions and $X_1,\ldots,X_n$ be independent and identically distributed ...
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2
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1
answer
52
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Jensen's inequality integral version for strongly convex functions
For a convex function $\phi$, not necessarily differentiable. If $\mu$ is a probability measure, and $f$ and $\phi(f)$ are integrable. We have:
$$
\phi \left( \int f \, d\mu \right) \leq \int \phi(f) \...
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47
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Jensen inequality with fraction expectation, determinant, and log function
I have a question when I try to have some analysis on the following expression.
\begin{align}
\mathbb{E}\left[\log_2\det\left(\mathbf{I}_L+\frac{1}{N_0+\sigma^2}\mathbf{X}\right)\right],
\end{align}
...
3
votes
1
answer
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Conditional Jensen's inequality proof correctness. Queries regarding convex functions.
Let $(Ω, \mathcal{F}, P)$ be a probability space and let $\mathcal{G} ⊂ \mathcal{F}$ be a sub-$σ$-algebra.
Conditional Jensen's inequality:
Let $φ : R → R$ be a convex function, $X$ and $φ(X)$ be ...
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30
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Jensen's inequality with expectation of log and determinant
I have a question when I try to have some analysis on the following expression.
\begin{align}
\mathbb{E}\left[\log_2\det\left(\mathbf{I}_L+\mathbf{X}\right)\right],
\end{align}
where $\mathbb{E}$ ...
0
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0
answers
28
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Single inflection point theorem
Let $f$ be a twice differentiable function on $\mathbb{R}$ with a single inflection point, let $S$ be a fixed real number and let $$g(x) = f(x) + (n-1)f(\frac{S-x}{n-1}).$$
If $x_1, x_2,...,x_n$ are ...
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Maximizing the sum of side lengths of a circumscribed tetrahedron
The original problem posed was the following:
On a sphere of radius $1$ are given four points $A,B,C,D$ such that
$$AB\cdot AC\cdot AD \cdot BC \cdot BD \cdot CD = \frac{2^9}{3^3}$$
I am aware of a ...
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1
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Let $f(x)=x^2+3x-3,x\gt0.$ If $n$ points $x_1,x_2,...,x_n$ are chosen on the x-axis, evaluate $\frac{x_1+x_2+...+x_n}n$
Question:
Let $f(x)=x^2+3x-3,x\gt0.$ If $n$ points $x_1,x_2,...,x_n$ are so chosen on the x-axis such that
(i) $\frac1n\sum_{i=1}^nf^{-1}(x_i)=f(\frac1n\sum_{i=1}^nx_i)$
(ii) $\sum_{i=1}^nf^{-1}(x_i)=\...
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Strict Jensen inequality for Ergodic Markov processes
Let $\vartheta(t)$ be a real Ergodic stationary Markov process solution to the stochastic differential equation
$$
\mathrm{d}\vartheta(t)=A_1(\vartheta(t),t)\mathrm{d}t+A_2(\vartheta(t),t)\mathrm{d}W(...
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votes
2
answers
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Bounding "Jensen's Gap": Elementary Approaches
The point of this post is to explore some "elementary" but general ways one can quantify the "gap" in Jensen's inequality. Specifically, let $h:\mathbb R\rightarrow\mathbb R$ be a ...
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Jensen's inequality for Euclidean spaces with extended reals
Let $X$ be a convex subset of $\mathbb{R}^n$, and let $p$ be a probability density function on $X$ (i.e. $\int_X p(x) dx = 1$), let $\phi:X\to \mathbb{R}\cup\{+\infty\}$ be a convex function with ...
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$\sum\limits_{\mathrm{cyc}} \frac{a}{2a^2+a+1}\leq\frac{3}{4}$, with $a,b,c\in\mathbb{R}^+,\ a+b+c=3$ [duplicate]
I am trying to prove the inequality:
$\sum\limits_{\mathrm{cyc}} \frac{a}{2a^2+a+1}\leq\frac{3}{4}$, with $a,b,c\in\mathbb{R}^+,\ a+b+c=3$.
Now if we consider the function $f:(0,+\infty)\to\mathbb{R},\...
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1
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Expectation of transformed mean of multiple random variables <= mean of the expectation of their transformation
I am trying to understand the following. If $M$ is the number of different random variables $X_1 ... X_M$, are there any conditions under which we can claim that:
$$E\left[f(\sum_{m=1}^M w_m X_m)\...
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1
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Show that $\mathbb{E}\left(|X|^q\right) \leq \left[\mathbb{E}\left(|X|^p\right)\right]^{\frac{q}{p}}$
Let $X$ be a random variable and let $p \in (0, \infty)$ such that $\mathbb{E}\left(|X|^p\right) < \infty$. Show that for all $q \in (0, p)$, we have $\mathbb{E}\left(|X|^q\right) \leq \left[\...
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Show inequality about riemann integrals with jensen inequality
I am having trouble proving the following inequality:
Let $f,h: [0,1] \to \mathbb{R}$ continuous, $h >0$ and $\phi: \mathbb{R} \to \mathbb{R}$ continuous and convex. Show that:
$$
\phi(\frac{\int^1 ...
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votes
1
answer
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Popoviciu's inequality
If $f(x): \mathbb R \to \mathbb R$ is a convex function, prove that
$$f(x) + f(y) + f(z) + 3 f(\frac{x + y + z}{3}) \geq 2 f(\frac{x + y}{2}) + 2 f(\frac{x + z}{2}) + 2 f(\frac{y + z}{2})$$
I proved ...
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votes
1
answer
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About Jensen’s inequality and Fano’s inequality [closed]
Let $X, X^′$ be independent with $X ∼ p(x)$, and $X^′ ∼ q(x)$, $x,x^′ \in \mathcal{X}$ . Then:
1.$Pr[X = X^′] ≥ 2^{−H(p)−D(p||q)}$,
2.$Pr[X = X^′] ≥ 2^{−H(q)−D(q||p)}$.
I don't kown how to prove 1 and ...
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Lower bounding the mean of a quadratic form with a positive semi-definite matrix
Let $\mathbf{X}\in\mathbb{R}^n$ be a vector random variable, and let $\mathbf{P}$ be an $n\times n$ positive semi-definite matrix.
I'm interested in deriving lower bounds on the expected value of the ...
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43
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Brezis' Exercise 4.9
I'm trying to solve below exercise in Brezis' Functional Analysis, i.e.,
Let $(\Omega, \mathcal F, \mu)$ be a measure space with $\mu(\Omega) < \infty$. Let $J:\mathbb R \to (-\infty, +\infty]$ be ...
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Convex function with a parameter H
Let us take $s_1, s_2, t_1, t_2 >0$ such that $ s_1 < t_1 < s_2 < t_2 $. Let us also assume that $H \in (0 , 1)$ and define
$$ a_1 = t_2 - s_1, \quad a_2 = t_2 - t_1, \quad b_1 = s_2 - s_1,...
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2
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Relation between $g(\mathbb{E}[X])$ and $\mathbb{E}[g(X)]$ when $g(X)=\frac{X}{1-X}$ is the "odds" function
Some context: Let's say we draw from an urn containing red and blue balls. We start at $n=1$, draw 1 ball, look at the color, and put it back, and repeat the process at $n=2$, $n=3$, etc., and every ...
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Pigeonhole coloring bound
I am stuck on this problem from a combinatorics book. I tried applying the pigeonhole principle to show that there are at least n/r numbers with the same color, and applied the formula for (x choose 2)...
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Show $2F\left(\frac{x+y}{2}\right)\geq F(x)+F(y)$
It's a follow up of Prove that $f(x,y)+f(y,z)\ge f(x,z)$ where $f(x,y)=\sqrt{x\ln x+y\ln y-(x+y)\ln(\frac{x+y}2)}$ :
I don't give the details but I show using second derivative :
if we have for $a,b,c,...
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Help me with this probability theory inequality---I have been stuck for weeks
I'm a bit stuck on a proof that seems to hold when I test in in graphing calculators, would love to hear some tips on strategies I might use.
Suppose we have two finite sets $\Omega$ and $\Phi$, with ...
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A probabilistic proof of Oppenheim's inequality?
Oppenheim's inequality is a standard result about the Hadamard product of positive definite matrices. It goes as follows, let $A=(a_{ij})_{i,j\leq n},B=(b_{ij})_{i,j\leq n} \in S_n^{++}$ where $S_n^{++...
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votes
2
answers
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Prove that $\prod_{i=1}^n\left(\frac{1}{x_i^2}-1\right)\ge (n^2-1)^n$
Let $n$ be a positive integer and $x_1,...,x_n$ positive reals such that $x_1+\dots+x_n=1$. Prove that $$\prod_{i=1}^n\left(\frac{1}{x_i^2}-1\right)\ge (n^2-1)^n.$$
Now notice that if we apply Jensen'...
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Show basic inequality for complex random variables
Suppose I have $n$ (not independent) complex random variables $X_i,i=1,\ldots,n$. I want to show the following
\begin{equation}
\mathbb{E}\left[\left|\sum_{j=1}^nX_j\right|^2\right]^{1/2}\leq \sum_{j=...
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1
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Reversal of inequality sign in Jensen's inequality
I am given a set of coefficients such that the affine combination $\{x_1, x_2, ..., x_n \} \notin conv(x_1, x_2, ..., x_n)$. How do I prove that under such given conditions the Jensen's inequality ...
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Inequality $\left(2-a-b\right)f\left(\frac{\left(2-a-b\right)^{2}-2dc}{2-a-b}\right)+af\left(a\right)+bf\left(b\right)\leq 16/25$
It's an inequality which refines the following question:
Prove or disprove that the inequality is valid if $x,y,z,u$ are positive numbers and $x+y+z+u=2$. :
Let $a,b,d\in[0,1]$ and $c\in[0.5,1]$ such ...
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Does the concavity of the log function justify the following inequality?
in an exercise solution I was presented with the following inequality and it's not making sense to me.
$ \frac{n}{n+1}\log{(\frac{1}{n}(x_1 + x_2 +...+ x_n))} +\frac{1}{n}\log{(x_{n+1})} $
$ \leq\log{(...
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JBMO-$2014$ Inequality question [duplicate]
Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$ {\left(a+\frac{1}{b}\right)^2}+{\left(b+\frac{1}{c}\right)^2} +{\left(c+\frac{1}{a}\right)^2}≥3(a+b+c+1)$$
My solution:
By Jensen'...
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3
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Prove that $ (a+b)^{2n} \leq 2^{2n-1}(a^{2n}+b^{2n})$ [duplicate]
I have to prove that for all $(a;b) \in \mathbb{R}^2$, and for all $n \in \mathbb{N}$ we have: $$ (a+b)^{2n} \leq 2^{2n-1}(a^{2n}+b^{2n})$$ without using induction.
I tried to use the convexity of $x^...
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Upper Bounding A Sum of Powers of Degrees in a Graph
Let $G$ be a simple, directed graph with $n$ vertices and $m$ edges. Let $V$ denote the vertex set of $G$. Given a vertex $v$ in $G$, let $\deg_{\text{out}}(v)$ denote its outdegree.
Finally, let $\...
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vote
1
answer
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A 'continuous version' of Jensen's inequality
Let $X$ denote a random variable that is smoothly distributed on $[0, 1]$ with PDF $f$. Consider
$$g(c) = \mathbb{P}(X< c) \phi(\mathbb{E}[X|X < c]) + \int_c^1 \phi(x)f(x) dx$$
where $\phi$ is ...
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Prove that $ \frac{x^2}{1+x} + \frac{y^2}{1+y} + \frac{z^2}{1+z} \geq \frac{1}{2} $
Assume that positive numbers a, b, c, x, y, z satisfy $cy+bz =a;
az + cx = b$$bx + ay = c$.
Prove that $ \frac{x^2}{1+x} + \frac{y^2}{1+y} + \frac{z^2}{1+z} \geq \frac{1}{2} $
I've tried appling ...
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4
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Prove that $\sqrt{a+ c} + \sqrt{b + c} \le f(x) \le 2\sqrt{\frac{a+ b}{2}+c}$, $f(x)=\sqrt{a\sin^2(x) +b\cos^2(x)+c} +\sqrt{a\cos^2(x)+b\sin^2(x)+c}$ [closed]
$f(x) = \sqrt{a\sin^2(x) + b\cos^2(x) + c} + \sqrt{a\cos^2(x) + b\sin^2(x) + c}$
a, b, c are positive real numbers
I turned $f(x)$ into
$\sqrt{a + c + (b - a)\cos^2(x)} + \sqrt{b + c + (a - b)\cos^2(x)...
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votes
1
answer
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What's the relation between the expectation of p-norm squared of a random vector before and after centering?
Let $a$ be a random vector with $\mathbb{E} a = b$ and $\mathbb{E} \|a\|_p^2 =\sigma^2$, where $1\leq p \leq \infty$. Is it true that $\mathbb{E} \|a-b\|_p^2 \leq 2\sigma^2$?
It is clear to me how to ...
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Application of Jensen Inequality for lower bound
In this work Jensen inequality is applied to turn Eq (11) to Eq. (12).
The essential part of Equation 11:
$L = \int q_{X_0}(x_{0}) log (\int q_{X_1,…,X_T | X_0}(x_1,…,x_T|x_{0})*C(x_1,…x_T) d_{x_1},…...
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Counterexample for Jensen's inequality for rank one convex function
We all know the Jensen's inequality is for all probability measures $\mu$ and all convex $h:\mathbb{R}^N\to\mathbb{R}$ it holds that
\begin{equation}
h\left(\int Xd\mu(X)\right)\leq \int h(X)d\mu(X).
\...
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1
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78
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Jensen's inequality but with geometric mean
If the inequality $f({\sqrt x}{\sqrt y})⩽{\sqrt f(x)}{\sqrt f(y)}$ is satisfied for all non negative $x,y$ in the domain, what can we say about the convexity of $f$ ? Or are there any other properties ...
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$ \mathbb P[X > 0] \geq \frac{\mathbb E [X]^ 2}{ \mathbb E [X^2]}$ if the mean is positive and finite
Let $Χ$ be a random variable for witch $0 < E[X] < \infty$. Prove
that: $ \mathbb P[X > 0] \geq \frac{\mathbb E [X]^ 2}{ \mathbb E
[X^2]}$
Hint: Use the fact that: $(\mathbb{1}_{\{X>0\}}\...
0
votes
0
answers
40
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Projection in Convex Optimization
I have fixed problem a and b, in (a) I use a counter method to prove the correctness of it, and with the conclusion if (a), (b) is easy to prove. But here comes problem c, here Δ stand for Bregman ...
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Bounding a double expectation of the entropy function
Consider a sequence of real numbers $0 \leq a_1 \leq \cdots \leq a_n \leq 1$ with $1/n\sum_i [a_i] = \mu$. Suppose further that $H(2 \mu - \mu^2) \geq H(\mu)$ (or equivalenly $\mu \leq \frac{3 + \sqrt{...
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2
answers
58
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Is $C^2$ necessary for convexity $\implies$ $f''(x) \geq 0$?
Let $f$ be a twice differential function defined on open interval $(a,b) \subseteq R$. Let $[c,d] \subseteq (a,b)$. I am able to prove that if $f''(x) \geq 0$ on $[c, d]$ then $f$ is necessarily ...
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1
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Jensen's for continuous convex combinations
Let $d_p(u(x), v(x)) = \left( \int_0^m \left| u(x)-v(x) \right|^p dx \right)^{1/p}$ be the distance according to the $\ell_p$ norm of two functions $u,v:[0,m]\to [0,M]$. Is the following true for the ...
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Any reference on Jensen inequality for measurable convex functions on a Banach space?
The only proof of Jensen inequality (and most general version) that I know is a direct consequence of the Fenchel-Moreau Theorem : If $X$ is a locally convex Hausdorff topological space, let $\mu$ be ...
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