Questions tagged [jensen-inequality]
For questions about proving and manipulating the AM-GM inequality. To be used necessarily with the [inequality] tag.
355
questions
2
votes
1answer
34 views
Contraction of Bellman Operator under general $L_p$ norms
We know that the Bellman Operator
$$
TV(s) = \max_a r(s,a) + \sum_{s' \in S}p(s'|s,a)V(s')
$$
is a contraction under $L_\infty$ norm.For reference one can see the following link Proof that Bellman ...
2
votes
1answer
34 views
Equality in Problem $2.12$, Rudin's RCA - Show that $h = \int_{\Omega} h\, d\mu$ a.e.
Suppose $\mu(\Omega) = 1$, and $h: \Omega\to [0,\infty]$ is measurable. If $A = \int_\Omega h\, d\mu$, prove that $$\sqrt{1+A^2} \le \int_\Omega \sqrt{1+h^2}\, d\mu$$ and that equality holds iff $h = ...
2
votes
1answer
111 views
Showing the data processing inequality
Let $X \sim P_\theta$ for some distribution $P_\theta$ parametrized by $\theta \in \Theta \subset \mathbb R$ and $Y \sim Q(\cdot | X)$ for some distribution $Q$. Assume that $P_\theta$ has a density $...
2
votes
1answer
33 views
Is there a “reverse” Jensen's inequality up to a constant?
Specifically I was wondering if for fixed $0<p<1$ there exists a constant $C$ such that $(E|X|)^p\leq C_pE(|X|^p)$ for any random variable $X$, giving an inverse of Jensen's inequality. From ...
0
votes
0answers
62 views
Relating Fatou's Lemma and Jensen's Inequality
If there is a sense in which $\liminf$ is a "concave" function, then we would expect Fatou's Lemma as a consequence of Jensen's inequality: Is there any way to make this precise?
For a small ...
-1
votes
1answer
35 views
Proof involving Jensen's inequality on random variables [closed]
Is it possible to use the Jensen's inequality to state the following:
$$E\left(\frac{1}{aX+b}\right)> \frac{1}{E(aX+b)}$$
where $X$ is a random variable and $a$ and $b$ are constants?
If not, is ...
1
vote
2answers
25 views
Is $E(E(Y|Z)|X)^2 \le E(E(Y|Z)^2|X))$ correct? (Jensen'sinequality)
I know that Jensen's inequality states that:
$\varphi(E(V)) \le E(\varphi(V))$, where $E$ stand for expected value, $V$ for a random variable and $\varphi$ must be a convex function.
Let $X$, $Y$ and $...
1
vote
2answers
49 views
Strict (or lack therefore) concavity of square root function
For the square root function $g(x) = \sqrt{x}$, we have
\begin{align}
g'(x) = 0.5x^{-0.5} \\
g''(x) = -0.25x^{-1.5} \\
\end{align}
So the second derivative is negative everywhere except if $x =...
2
votes
1answer
43 views
Multivariate Jensen's inequality in a probabilistic setting: When does equality hold?
I am having trouble understanding the concept of Jensen's inequality in a probabilistic setting with multiple variables. Specifically, I am interested in cases where equality holds. I understand that ...
-2
votes
1answer
36 views
Jensen Inequality and Expected value
I am trying to proof that:
Let f $\in$ $C^2(R,R)$,$f''>0$, and $E[f(X)]=f(E[X])$.
I am trying to proof that X should be a constant.
I know that $f(E[X])\leq E[f(X)]$, this holds because of the
...
1
vote
0answers
52 views
proof verification: Jensen's inequality
Following is the Jensen's inequality in Rudin's "Real and Complex Analysis".(Thm 3.3)
Let $\mu$ be a positive measure on a $\sigma-$algebra $\mathfrak M$ in a set $\Omega$, where $\mu(\...
2
votes
0answers
76 views
Showing a convolution is Lipschitz. In measure.
Let $\Lambda:\mathbb{R}^d\to \mathbb{R}^d$, be Lipschitz. Is the convolution of $\Lambda$ with a measure Lipshchitz? That is do we have
$$
\int_{\mathbb{R}^d} \| \Lambda*\rho(x) - \Lambda*\mu(x)\|^2 ~ ...
0
votes
0answers
23 views
Jensen inequality cross entropy
I don't remember how to do math
Prove that $\sum_{j=1}^{k} r_{j}{\log p_{j}} -\sum_{j=1}^{k} r_{j}{\log r_{j}} <=0$
Can I just put all expressions in one sum symbol, use log probability $\log(p_{j}...
1
vote
1answer
52 views
Show that $E_f(\log f)\ge E_f (\log g)$
Let $f(x), g(x)$ be probability densities defined on $R^n$ with $f(x), g(x)>0$ for all $x$. Show that $E_f(\log f(x))\ge E_f (\log g(x))$
with $E_f(h(x))=\int_{-\infty}^\infty h(x) f(x) dx$ is the ...
0
votes
1answer
29 views
Jensen inequality and mixed risk portfolio
Let $X$ be a random variable indicating the gains in some investment, let $\mu = E[X]$ be the expected value, and $U(X)$ is the utility obtained from that gain.
If $u$ is convex, then $U^{\prime \...
1
vote
2answers
38 views
Is the integral over a convex mapping of the solution to the heat equation monotonic?
I have the following problem:
Suppose $u = u(t,x)$ solves the Cauchy-Problem for the (one-dimensional) heat equation: $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$$ $$u(0,x) = g(...
1
vote
1answer
56 views
Use Jensen's inequality to show that $e^{\frac{\sum^3_{i=1} X_i}{3}}$ is not an unbiased estimator of $e^{\theta}$
My problem:
Let $X_{1}, X_{2},X_{3}$ be I.I.D. Poisson variables with common mean $\theta > 0$.
Use Jensen's inequality to show that $e^{\frac{\sum^3_{i=1}X_i}{3}}$ is not an unbiased estimator of $...
0
votes
1answer
19 views
An inequality in expected values of Random Variable with different PDFs
Prove that $E \ [log \ f(X)] \ge E \ [log \ q(X)]$, if X is a random variable with PDF $f(x)$ and $q(x)$ is any valid PDF.
Hint provided: Use Jensen's Inequality.
This means: $E\ [log \ f(X)] = \int f(...
3
votes
2answers
60 views
Proving Jensen's inequality for the general case starting from the finite case
For a finite convex combination, Jensen’s inequality is given by: $$f\left(\sum_i^n a_i x_i\right) \leq \sum_i^n a_i f(x_i)$$ for a convex $f$. Proving this is not so bad starting from the definition ...
4
votes
0answers
42 views
Prove that $c_V(X) \geq c_V(f(X))$ for $X>0$ and $f$ concave, positive, increasing,
I have a square-integrable random variable $X > 0$. Function $f : \mathbb{R}^+ \rightarrow \mathbb{R}^+ $ is concave, monotonically increasing and $f(0) = 0$.
I would like to prove that (and make ...
0
votes
2answers
64 views
How to remember which direction Jensen's inequality is in?
For a real valued random variable and a convex function $f$, Jensen's inequality is given by:
$$f(\mathbb E X) \leq \mathbb E f(X)$$
How do you remember which direction the inequality is in? I always ...
1
vote
1answer
60 views
On the requirements of the Jensen inequality for conditional expectation
I'm often reading this version of the Jensen inequality for conditional expectation:
Let $(\Omega,\mathcal A, P)$ a probability space and $X$ a integrable random variable. Then for any convex ...
0
votes
0answers
66 views
Proof that function larger than its bounds
Setup. Let $f_k(x) = 1-\exp\left(-\frac{xk+(1-x)(k+1)}r\right)$ and $g_k(x) = xf_k(x)^k + (1-x)f_k(x)^{k+1}$. The domains are $x \in (0, 1)$, $k \in \mathbb{N}$ and $r \in \mathbb{R}$.
An intuitive ...
5
votes
2answers
118 views
Alternative approaches to prove the following inequality
For $a,b,c \in \mathbb{R^+},$ prove that
$$\left(\dfrac{2a}{b+c} \right)^{\frac{2}{3}} + \left(\dfrac{2b}{c+a}\right)^{\frac{2}{3}} + \left(\dfrac{2c}{a+b}\right)^{\frac{2}{3}} \geq 3.$$
I managed to ...
1
vote
0answers
56 views
Using Cauchy-Schwarz and Jensen's inequality
I need a certain estimation from a paper in which it has been described to use the Cauchy-Schwarz inequality and Jensen's inequality. I did some calculations and had a few tries but I don't seem to ...
0
votes
1answer
40 views
Applying Jensen's inequality to a convex tarsnformation of a non-convex simple random variable
My objective is to (numerically) maximize an expectation of the form of
$$
E[\exp(g(\mathbf{X}))]
$$ where $\mathbf{X}$ is random variable and $g$ is a non-convex function on $\mathbf{X}$. If I ...
2
votes
1answer
97 views
Equality in Jensen's inequality for non convex functions
Jensen's inequality, $\mathbb{E}[\phi(X)]\geq\phi(\mathbb{E}[X])$, holds for any convex function $\phi$ and random variable $X$.
Given that $\mathbb{E}[\phi(X)]=\phi(\mathbb{E}[X])$ for some non-...
0
votes
1answer
134 views
Double refinement of Nesbitt's inequality
It's a double refinement of Nesbitt's inequality :
Claim :Prove or disprove this statement.
Let $a,b,c>0$ such that $1\leq c\leq b\leq 2$ then we have :
$$\frac{a}{\sqrt[3]{4(b^3+c^3)}}+\frac{b}{a+...
3
votes
1answer
154 views
A Polynomial-Inequality problem from Vietnam National Olympiad 2021
Abstract
I have just completed 2 days of our National Olympiad. This year's problems are not difficult, yet new and strange to many of the students. Mentioned below will be problem 5 out of 7, the ...
1
vote
1answer
87 views
Jensen Inequality Euclidean Norm
I'm trying to prove $$\mathbb{E}\|X\|\geq \|\mathbb EX\| $$ for any random matrix $X$ and the spectral norm $\|\cdot \|$. To finish my proof all I need is the same statement for the euclidean norm $|\...
0
votes
1answer
40 views
Dominated convergence theorem by Jensen inequality
Define functional $\psi(x_n):=\limsup_n x_n$ on a sequence of positive numbers $x_1,x_2...$. We can check that $\psi$ is convex.
Then if $f_n \to f$ pointwise, by Jensen's inequality,
$$\limsup_n \int ...
1
vote
0answers
59 views
Expected value of integral with random variable on bound
I'm trying to write the following as a single integral, however I did not get the decired results yet, and tried searching this site without succes.
What I'm trying to rewrite is:
$$\mathbb{E}\int^...
3
votes
0answers
162 views
Refinement of $\left(\frac{a+1}{a+b} \right)^{\frac25}+\left(\frac{b+1}{b+c} \right)^{\frac25}+\left(\frac{c+1}{c+a} \right)^{\frac25} \geqslant 3$
It's a refinement of Prove $\left(\frac{a+1}{a+b} \right)^{\frac25}+\left(\frac{b+1}{b+c} \right)^{\frac25}+\left(\frac{c+1}{c+a} \right)^{\frac25} \geqslant 3$ .
It's a attempt to find a simple proof ...
4
votes
1answer
127 views
Show the inequality : $a^{(2(1-a))}+b^{(2(1-b))}+c^{(2(1-c))}+c\leq 1$
Claim :
Let $0.5\geq a \geq b \geq 0.25\geq c\geq 0$ such that $a+b+c=1$ then we have :
$$a^{(2(1-a))}+b^{(2(1-b))}+c^{(2(1-c))}+c\leq 1$$
To prove it I have tried Bernoulli's inequality .
For $0\leq ...
0
votes
0answers
30 views
Is there a simpler way to prove that a fractional function with three variables is convex?
How to prove that f(x) is convex?
For all $$x_i>0$$
we have
$$ f(x) = \frac{1}{x_1 - \frac{1}{ x_2 - \frac{1}{x_3} }} $$
I tried to take the second derivative (hessian matrix) but it was too long ...
1
vote
1answer
35 views
Some converse of Jensen inequality for some extreme random variable
This problem arises in my research in a more general form. It concerns information theory and statistical bounds using information theory. I tagged this question with Jensen inequality since the ...
2
votes
0answers
127 views
Existence of an asymptote for $g(x)=\frac{f(x)f'(x)+f(1)f'(1)}{f'(x)+f'(1)}-f\left(\frac{xf'(x)+f'(1)}{f'(x)+f'(1)}\right)$
Working with the Slater's inequality (companion of Jensen's inequality) I find this statement :
Let $f(x)$ be a continuous,$n$ times differentiable ,convex and non constant on $(0,\infty)$ and ...
0
votes
1answer
33 views
Verify that $M_{X}(t) \ge e^{t\mu}$ using Jensen's identity
Suppose that a real valued random variable with a probability density of $$f(x) = \begin{cases}
\frac{1}{8}(x+1) &:-1 <x<5\\
0 &: \text{else}
\end{cases}$$.
I need to verify that $M_{X}(...
6
votes
1answer
209 views
Probability of a deviation when Jensen’s inequality is almost tight
Let $X>0$ be a random variable. Suppose that we knew that for some $\epsilon \geq 0$,
\begin{eqnarray}
\log(E[X]) \leq E[\log(X)] + \epsilon
\tag{1} \label{eq:primary}
\end{eqnarray}
The question ...
2
votes
1answer
154 views
Prove this refinement of Nesbitt's inequality based on another
Let $a,b,c\in[1,2]$ such that $a,b$ are constants then prove :
$$f(c)=\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{b+a}\geq h(c)=(c-1)\frac{g(2)-g(1)}{2-1}+g(1)\geq g(c)=\sqrt{\frac{9}{4}+\frac{3}{2}\frac{(a-...
1
vote
2answers
85 views
Stronger than Nesbitt's inequality using convexity and functions
Hi it's a refinement of Nesbitt's inequality and for that, we introduce the function :
$$f(x)=\frac{x}{a+b}+\frac{b}{x+a}+\frac{a}{b+x}$$
With $a,b,x>0$
Due to homogeneity we assume $a+b=1$ and we ...
1
vote
0answers
31 views
prove for a maximal inequality using Jensen's inequality
Problem
Let $\{Z_i\}_{i \in \{1,\dots, n\}}$ be a family of independent random variables and $v>0$. Given that
$$ \forall i \in \{1,\dots,n\}, \lambda > 0: \mathbb{E}[e^{\lambda Z_i}] \leq e^{\...
0
votes
1answer
35 views
Convexity in NMF proof
I am currently studying a proof for the NMF multiplicative update rule based on generalized Kullback-Leibler divergence in the paper Algorithms for Non-negative Matrix
Factorization, and for some ...
-2
votes
1answer
57 views
Inequality between geometric mean and harmonic mean [closed]
Suppose that $x_1,\dots,x_n>0$. Prove that $$\sqrt[n]{x_1x_2\dots x_n}\geq \dfrac{n}{\frac{1}{x_1}+\dots+\frac{1}{x_n}}$$ by using Jensen's inequality for some suitable function $f$.
0
votes
0answers
34 views
Ask for a proof that expectation falls in convex domain
I am stuck at the first sentence of the following proof of Jensen’s inequality, appearing in Erhan Çınlar's text "Probability and Stochastics", P70-71:
I am pretty frustrated not only by my ...
0
votes
1answer
14 views
Upperbound of Kullback-Leibler between marginals
I want to know if for every pair of $p(x, y)$ and $q(x, y)$ the following inequality
$$
\int_X p(x) \log \frac{p(x)}{q(x)} dx \leq \int_X\int_Y p(x, y) \log \frac{p(x, y)}{q(x, y)}dxdy
$$
is valid.
I ...
0
votes
1answer
65 views
Applying Jensen's inequality to find a bound
I'm not sure how to begin with this question, apart from writing some basic relationships based on Jensen's inequality. Any suggestions would be appreciated.
For arbitrary α and β satisfying 0 < α ...
0
votes
1answer
115 views
Jensen's Inequality
For arbitrary α and β satisfying 0 < α ≤ β, use Jensen’s inequality to find a bound of the form
E[|X|$^α$] ≤ f(E[|X|$^β$]) for some function f.
Doesn't this bound follow naturally from the fact ...
2
votes
1answer
40 views
Undertanding Jensen's inequality proof on countable basis Hilbert with Radon measure
Given the following proposition :
$\textbf{Proposition :}$ $X$ Hilbert space with countable basis, $\mu$ Radon measure on $X$, $\phi \in L_{\mu}^{1}(X)$ and $f \in \left\lbrace f : X \longmapsto \...
0
votes
0answers
117 views
Finding bounds for the absolute value of function raised to positive constant
I am trying to show the below inequality
$$|x|^{a_1} + |y|^{a_2} \leq C(|x-y|^{a_1} + |y|^{\max(a_1,a_2)}) $$
for constants ($a_1>0,a_2$>0) and $C$ is some other constant. What is the quickest ...
