Questions tagged [measure-theory]

A field of mathematics concerning measures, which are functions mapping sets to real numbers that generalize the idea of area and volume. Measure theory underlies modern treatments of probability. Questions about measure theory with a weak connection to probability or statistics may be more suitable for math.stackexchange.com than this site.

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Measuring distance between distributions with different kernels functions

currently I am using maximum mean discrepancy to measure distance between multivariate distributions. The related reference is like An explicit description of the reproducing kernel Hilbert spaces of ...
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Is regression using Gaussian processes a special case of Bayesian inference?

The standard formulation of gaussian process (GP) regression goes as follows. Suppose the prior distribution is $f \sim \mathcal{GP} (0, k)$, and a sample $S$ of size $n$ is drawn, \begin{equation} S =...
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Chain rule for KL divergence, conditional measures

The chain rule for KL divergence is widely seen in the theoretical machine learning literature and generally referenced to [1, Theorem 2.5.3]: $$ \text{KL}[p(x, y) \mid q(x, y)]= \text{KL}[p(x) \mid q(...
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clear example of sigma algebra generated by r.v [duplicate]

I can't understand how a sigma algebra generated by random variable. can you more explain and give me some examples. for example in coin toss we have a sample space and sigma algebra such that it's ...
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Relationship between deterministc function of random variables

Given a discrete $P(X,Y,Z)$ let's call $\Omega$ the set of all deterministic functions $f: XYZ \rightarrow W$ and $\Omega'$ the set of all deterministic functions $f': XY \rightarrow V$. Is it correct ...
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Show $X_n \to 0$ in probability

I am asked to show : Let $X$ be a real-valued random variable on $(\Omega, F , P)$ and define $X_n(\omega) = nX(\omega)$ if $n<X(\omega)\le n+1$ and $0$ if else. Prove that $X_n \to 0$ in ...
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The book Probability Theory by Klenke, Exercise 1.2.1, I wonder if it’s an error?

The book Probability Theory by Klenke has for Exercise 1.2.1 the following: Let $$A =\{(a,b]\cap \mathbb Q : \ a,b\in\mathbb R, \ a\le b\}.$$ Define $$\mu: A \to [0,\infty)$$ by $$ \mu((a,b]\cap \...
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Does the law of total probability apply to hazards?

Consider the hazard function for a random variable $T$, conditional on some other random variable $U$: $$ h(t|U=u)=\lim_{\Delta t\rightarrow0}\frac{P(t<T<t+\Delta t|T>t,U=u)}{\Delta t} $$ ...
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Is $\left\{(\mathbf{X_n}^T\mathbf{X_n}/n)^{-1}\right\}_{n=1}^\infty$ uniformly integrable (UI)? What assumptions make it UI?

$\left\{(\mathbf{X_n}^T\mathbf{X_n}/n)^{-1}\right\}_{n=1}^\infty$ Let $\mathbf{X}_n$ be the usual data matrix in standard multiple regression where I have used the subscript $n$ to indicate the number ...
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Is it possible to interchange the quantile operator and a measurable monotone function? $Q_\theta(f(X)) = f(Q_\theta(X))$

Let $Q_\theta(X)$ is the $\theta^{th}$ quantile of a random variable $X$, and if $f$ is a measurable strictly increasing function. I want to know if $Q_\theta(f(X)) = f(Q_\theta(X))$. I know that for ...
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What's the term for a r.v. X “upper bounding” Y probabilistically? [duplicate]

What's the term for when a random variable $X$ has a higher probability of being greater than t than the probability of $Y$ being greater than $t$ for all real $t$? I recall reading a term for this ...
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Show that maximum of two random variables is a random variable

If we have that $X$ and $Y$ are random variables, how do we prove that $Z=max(X,Y)$ is also a random variable ? I want to do this by showing that $Z$ is measurable, but I don't know how to do this.
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What is the proper (measure theoretic) definition of a random variable?

(1) If we believe the following link: http://www.columbia.edu/~md3405/DT_Risk_2_15.pdf Then a random variable is a map from a probability/measurable space to a metric space, i.e. (2) If we believe ...
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Intermediate proof for Glivenko-Cantelli Theorem

Show that for any cdf, the following holds: $sup_{x\in \mathbb{R}}|F_n(x)-F(x)|\leq sup_{u\in [0,1]}|\overline{F}_n(u)-F(u)|$ Where $F_n:=\frac{1}{n}\sum_{i=1}^n1_{(-\infty,x]}(X_i)$ is the empirical ...
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Probability of intersection involving a continuum

I'm trying to derive pascal's rule of succession that I saw on a 3b1b video in a rigorous-like way(i don't have a solid background in measure theory so I can't pretend to be rigorous). let $ ω=[0,1] × ...
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How to understand conditional expectation w.r.t sigma-algebra: is the conditional expectation unique in this example?

This board has many questions on how to understand conditional expectations w.r.t a $\sigma$-algebra; it's clearly a topic that confuses many. I am one of them. I read all the other questions and ...
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Covariance and indicator function relations

Let $\mathcal A$ and $\mathcal B$ be two (sub) sigma-algebras of some probability space. Rio (2017), pp. 4 defines the coefficient: $$\alpha(\mathcal A,\mathcal B)=2\sup\{\lvert Cov(1_A,1_B):(A,B)\in\...
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Product of two probability density function

Suppose $f$ and $g$ are two probability density functions. I have seen economists use $\int f(x)g(x) dx$ as some kind of similarity measure. For example, Jaffe (1986) uses sum of product of two ...
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Independence of a statistic from size of sample

If you have some (non-constant) statistic $T(X_1, X_2,...,X_n) = f(X_1,X_2,...,X_n)$, is it independent of the number of elements in the sample $(X_1, X_2, ..., X_n)$? That is, is it independent of $n$...
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Difference between measure and metric in context to information and mutual information

In measure theory, a measure is intrinsic property of the set which informs about the size of the set. On the otherhand an information say a mutual information is a diminishing return property, that ...
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Obtaining an expression for empirical mean from empirical CDF definition

This is my first post so I will try to be as clear and concise as possible. I am doing a course in statistics and we define the true mean of a random gaussian variable to be as follows: $\mu$ = $\int_{...
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Probability Measure for Continuous Random Variable

Say a continuous random variable $X \in \mathbb{R}$ has a probability density function (p.d.f.). Is it correct that, change the probability measure $\Rightarrow$ define a new p.d.f. for X ? In other ...
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Probability Assigned to Distance between Two People Random Walking in a Room [closed]

In a scenario where there are two people in the rooms next to each other randomly walking in a room I want to know if we can compute PDF of distance between the two people. So the way I tried to ...
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Formal book about GLM

I'm searching for a book who treat GLM in a formal way, with a measure theoretic approach. Someone could help me? i try to be more specific Suppose $ (\Omega,\mathcal{F},\mathbb{P})$ is a probability ...
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Lots of modes of convergence for random variables but why is there nothing on convergence of probability density functions?

There are numerous modes of convergence for random variables. But why do I never read anything about convergence of probability density functions? It seems like this would also be an important notion ...
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A function of random variables $X_1, …, X_k$ that goes from $\mathcal{R}^k$ to the reals is measurable with respect to $\sigma(X_1, …, X_k)$

I'm reading Resnick's "A probability Path" and doing exercise 3 on page 85. The statement is: Suppose $f : \mathcal{R}^k \rightarrow \mathcal{R}$ and $f \in \mathcal{B}(\mathcal{R}^k) / \...
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General Questions on Measure Theory [duplicate]

The notion I have received while trying to study Measure theory that it is helpful when we move from discrete domain to continuous domain because in discrete domain we can count individual entities ...
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A local base for space of probability measures with Prohorov metric [closed]

Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...
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This mixed-type family of random variables has no dominating measure, so a likelihood function can't be defined?

Let $$ X = \begin{cases}\theta & \text{with probability 1/2}\\ Z\sim N(0,1) & \text{with probability 1/2.} \end{cases}$$ Here, $\theta\in\mathbb{R}$ is the parameter to be estimated. It ...
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Why is measure theory needed to understand continuous random variables and probability density functions in particular?

Prefacing the question with the fact that I have no knowledge of measure theory. I would prefer a conceptual answer, as there already many mathematical ones. Also, why don't we need measure theory to ...
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How to intuitively visualize measure theory concepts via diagrams? [closed]

Visualizing Measure Theory via Diagrams / Drawings / Geometry Hi, I'm starting to learn about Measure Theory in order to study more advanced topics in Data Science. However, the explanations are ...
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$\inf$ of a sequcence of random variables bigger than some $a\in\mathbb{R}$

Suppose we have sequence of random variables $\{X_n\mid n\in\mathbb{N}\}$, defined on a probablity space $(\Omega,\mathcal{F},\mathbb{P})$. Then we define $(\inf_{n\in\mathbb{N}}X_n)(\omega)=\inf_{n\...
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Likelihood function when there is no common dominating measure?

When we have a statistical model $\{ P_\theta, \theta\in\Theta\}$ on some common probability space, usually we define "the" likelihood function $L(\theta)$ via a Radon-Nikodym derivative ...
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Sigma algebra generated by random variable on a set with generators

I'm having trouble proving an intuitive result I found in these lecture notes I'm using for self-study (1.2.14 there). Suppose $X$ is a $(\mathbb{S}, \mathcal{S})$-valued random variable (from $(\...
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What's the relationship between these two statements?

Suppose I have a set $A\subset R^2$ and an estimator for set $A$ denoted as $\widehat{A}$. Let $\widehat{A}\Delta A$ denote the symmetric difference between set $\widehat{A}$ and set $A$. For some ...
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Explain the descriptive statistics notion of population (distribution) to a measure theorist

My understanding of probability theory is a rigorous measure-theoretic one and I know almost nothing about descriptive statistics. However, I need to understand some of the commonly used notions in ...
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Integral of distribution followed by Bernoulli

I am new to learning probability theory. Here I got confused but I had a sense of feeling it's correct. This is what I saw from the lecture notes. Let $P\sim Bern(p), Q\sim Bern(q)$. Then is the ...
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When does it matter if random variables are defined on a common probability space?

If I've got two random variables $X$ and $Y$, when does it matter if I think of them both having the same sample space $\Omega$ or not? I sometimes hear phrases like "assuming a common ...
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De Finetti's Coherence Principle and Frequentist interpretation

So, without proof or citation, I often see that the Coherence Principle by de Finetti does not hold with Frequentist statistics. It is pretty easy to create examples of this fact. The exception ...
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Which definition of Kullback-Leibler-Divergence for discrete & continuos variables makes sense?

Motivation In variational inference, one tries to approximate a posterior density $p(\theta|y)$ by another,easier, density $q(\theta)$ in terms of the Kullback-Leibler-Divergence $$D_{KL}(q(\theta)||p(...
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Conditional probability density from probabilities

I am trying to understand conditional probablility densities in relation to the conditional probablilities. From the Measure-theoretic definition on Wikipedia, if $X$ and $Y$ are non-degenerate and ...
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Probability measure condition

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let $\{A_i\}_{i\in\mathbb{N}}\subseteq\mathcal{F}$ be a countable partition of $\Omega$, where $\mathbb{P}(A_i)=ab^i,\forall i\geq1$. ...
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What's the event space of a single coin toss?

Take a probability triple, $(\Omega,\mathscr{F},\mathbb{P})$, representing a single coin toss. Then \begin{align} & \Omega = \{H,T\}. \tag{Prop. 1} \\ \end{align} Now, \begin{align} \text{A } \...
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Optimize a Function with Respect to a Set

Suppose I have some set of points $X = \{x_1,...,x_N\}, x_i \in \mathbb{R}^d$ and a measure $\mu$ (i.e. a probability distribution) defined with respect to those points. Specifically, if $A \subset \...
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Applying Bayes' rule in a more technical way when densities don't exist

Say $y \mid x \sim \text{Normal}(Ax, B)$ and $x \sim \text{Normal}(c,D)$. Let's assume further that $y \in \mathbb{R}^1$ and $x \in \mathbb{R}^2$. To find $p(x \mid y)$ we can usually do \begin{...
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If likelihood is statistically sufficient, how could some cases have no sufficient statistics?

Almost all elementary texts clarify that in some cases minimal sufficient statistics might not exist. However, it seems that the likelihood itself induces a partition that essentially provides a ...
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Are Bhattacharyya coefficient and total variation distance complementary?

I was reading about total variation distance, and, as I understood it, it should measure how much two probability measures don't overlap. To be clear: in these images Bhattacharyya coefficient is ...
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Kullback-Leibler Divergence continuous for measures?

in one of the commments to this post concerning the application of Kullback-Leibler-divergence between measures that do not fulfill the necessary absolute continuity (e.g. point mass vs. continuous) , ...
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Reference request: Modulus of continuity of the map from parameter space to the space of probability measures

I am thinking of the following condition in my research: Let $\Theta \subset \mathbb R$ be the parameter space and $\mathbb P_\theta$ be the probability measure on sample space $\mathbb X$ ...
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Defining continuous random variables via uncountable sets

At several sources I have encountered the following two definitions of a continuous random variable associated with uncountable sets: a) uncountable range: The random variable X is continuous if its ...

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