# Questions tagged [inequality]

Use this tag if you question involves the use of an inequality. The inequality may have probabilistic origins or be a purely mathematical inequality. Do not use for measures of inequality, for instance income inequality. For that use the [diversity] tag.

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### REINFORCE algorithm, help for the proof of the variance reduction by subtracting a baseline

I'm trying to find a proof or an approximate argument justifying that, in the REINFORCE algorithm, subtracting a baseline to the episode reward reduces the variance. I believe this proof can be done ...
1 vote
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### If event $A$ is a union of elements of $S(A)$, then $\min_{Z\in S(A):Z\subset A}P(B\mid Z)\leq P(B\mid A)$ for any event $B$

Let $A$ and $B$ be events in a probability space and $S(A)$ a collection of events such that $A$ is a union of some elements in $S(A)$. How could I then conclude that the conditional probabilities ...
1 vote
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### Is there any theory pertaining to an assumption to would allow the following inequality to hold (constrained LASSO)

Suppose we observe the vector-matrix pair $(y,\mathbf{X})\in\mathbb{R}^n\times\mathbb{R}^{n\times d}$ which is linked by the observation model: y=\mathbf{X}\theta^*+\epsilon \end{...
155 views

### Is it true that $\langle X^4\rangle \ge 3 \langle X^2\rangle^2$?

Consider a real random variable $X$ with zero mean. Does the following inequality hold in general? $$\langle X^4\rangle \ge 3 \langle X^2\rangle^2$$ I'm not sure how to prove this or if a counter-...
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### Oracle Inequality : In basic terms

I'm going through a paper that uses oracle inequality to prove something but I'm unable to understand what it is even trying to do. When I searched online about 'Oracle Inequality', some sources ...
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### What is the correlation between a random variable and its probability integral transform?

Are there known bounds on the $\operatorname{cor}(X,F(X))$? $X$ is a random variable with CDF $F(X)$. Let $X$ have a fixed variance, for example $\operatorname{var}(X)=1$. What $X$ can maximize or ...
1 vote
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### Conditionalizing events on more than one event

I am currently working on a question which seems to have an obvious answer, but it it seems just impossible for me to find a stringent proof of this relation (if it is true). Imagine the following ...
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### Why is "Jensen's Inequality" Important in Probability?

Why is Jensen's Inequality Important in Probability and Statistics? I was reading the Wikipedia page on "Convex Functions" (https://en.wikipedia.org/wiki/Convex_function), and came across ...
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### Statistical Test for Some Inequality Condition

Suppose that we have three random variables, named $V_1, V_2$ and $V_3$. Here, I want to test the following inequality: $V_1 \geq V_2, V_3$ Is there any reference dealing with this topic?
257 views

### Problem understanding the intuition behind Slepian's inequality

Slepian's inequality is defined as follows: Let $X\in\mathbb{R}^n$ and $Y\in\mathbb{R}^n$ be centered Gaussian random vectors such that \begin{align} \mathbb{E}X_iX_j&\geq \mathbb{E}Y_iY_j,\quad \...
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1 vote
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### Holder's inequality in the case of $L_1$ and $L_{\infty}$ norm

I am referring to Wainwright's High-Dimensional Statistics book, where at some point it is deduced that \frac{w'X\Delta}{n}\leq \left\lVert\frac{w'X}{n}\right\rVert_{\infty}\lVert\...
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### Difference of two KL-divergence

The Kullback-Leibler (KL) divergence between two distributions $P$ and $Q$ is defined as $$\mbox{KL}(P \| Q) = \mathbb{E}_P\left[\ln \frac{\mbox{d}P}{\mbox{d}Q}\right].$$ My question is that suppose ...
It is common knowledge that: $$$$\label{3} Var(X) \geq 0$$$$ for every random variable $X$. Despite this, I do not remember seeing a formal proof of this. Is there a proof ...