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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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Multidimensional Stieltjes measure functions

In the first chapter of the book "Probability: Theory and Examples" by Rick Durrett, a function $F:\mathbb{R}^n\to\mathbb{R}$ is said to be a Stieltjes measure function if $F$ is non-decreasing, that ...
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Example of a non-measurable maximum likelihood estimator

If you have a measurable parameter space $(\Theta, \mathcal{F})$ and a parametric family of probability measures $(P_\theta)_{\theta \in \Theta}$ on a measurable space $(\mathcal{X}, \mathcal{B})$ ...
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The meaning of distance measure in Additive attention and Dot Product Attention

Currently, I am studying about Transformer (reading this paper; attention is all you need). In the paper, the author said, (3.2.1) Additive attention computes the compatibility function using a ...
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Clarification on the concept of a cumulative distribution function of a measure (measure theory)

I was asked to show that $g_f(x)=\mu(f\leq x)$ defines a cumulative distribution function for any measurable function $f$. Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $(\mathbb{R},\...
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$P(h\leq k)=P(t \leq k),\forall k\in \mathbb{R}$ implies $h=t$

Consider the measurable space $(A,\mathcal{A})$. Let $h,t:A\rightarrow \mathbb{R}$ be mesurable functions. Show that If $P(h\leq k)=P(t \leq k),\forall k\in \mathbb{R}, \forall P$ probability on $(A,\...
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Expectation of a constant (basic topic)

Consider the space $(\Omega,\mathcal{F},P)$. Show that $E(c)=c,\forall c\in \mathbb{R}$. I thought about two distinct ways to show that. $E(c)=\int_\Omega c\ dP=c\int_\Omega dP=cP(\Omega)=c$; Let $f=...
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Optimal proposal for the Metropolis-Hastings algorithm using Tierney's theorem

Let $(E,\mathcal E,\lambda)$ be a measure space $p:E\to[0,\infty)$ be $\mathcal E$-measurable with $$c:=\lambda p\in(0,\infty)$$ and $$\mu:=\underbrace{\frac1cp}_{=:\:\tilde p}\lambda$$ $q:E^2\to[0,\...
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Express the mean acceptance rate of the Metropolis-Hastings algorithm as a total variation distance

Let $(E,\mathcal E,\lambda)$ be a measure space $p:E\to[0,\infty)$ be $\mathcal E$-measurable with $$c:=\int p\:{\rm d}\lambda\in(0,\infty)$$ and $$\mu:=\underbrace{\frac1cp}_{=:\:\tilde p}\lambda$$ $...
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19 views

Express the mean acceptance ratio of the Metropolis-Hastings algorithm as a total variation distance

Short question in Theorem 1 on page 3 here https://openreview.net/pdf?id=Hkg313AcFX, he mean acceptance ratio is expressed in terms of a total variation distance. However, I don't understand the ...
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148 views

Expression for the mean acceptance rate of the Metropolis-Hastings algorithm

Let $(E,\mathcal E,\lambda)$ be a measure space $p:E\to[0,\infty)$ be $\mathcal E$-measurable with $$c:=\int p\:{\rm d}\lambda\in(0,\infty)$$ and $$\mu:=\underbrace{\frac1cp}_{=:\:\tilde p}\lambda$$ $...
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1answer
47 views

Is the transition kernel of a Metropolis-Hastings chain of the form $P(x,A)=\varrho(x)\tilde P(x,A)+(1-\varrho(x))1_A(x)$?

After equation (1) at page 3 of this paper it is claimed that the transition kernel of a Markov chain generated by the Metropolis-Hastings algorithm is of the form $$P(x,A)=\varrho(x)\tilde P(x,A)+(1-\...
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Unbiased estimator of exponential of measure of a set?

Suppose we have a (measurable and suitably well-behaved) set $S\subseteq B\subset\mathbb R^n$, where $B$ is compact. Moreover, suppose we can draw samples from the uniform distribution over $B$ wrt ...
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Non-metrics give “pathological” solutions: what does this mean?

In this set of slides on DTW, slide 25 says that we generally prefer metrics over measures because, "Non-Metrics can sometimes give pathological solutions when clustering or classifying data etc." ...
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Does this notation in the context of reinforcement learning have an unambiguous measure-theoretic interpretation?

In the context of reinforcement learning, I have seen the formula $$V^\pi(s)=\mathbb E_{\tau \sim \pi}[R(\tau)|s_0=s]$$ and $$V^\pi(s)=\mathbb E_{a\sim \pi}[Q^\pi(s,a)]$$ Does this notation have ...
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Can a discrete random variable be absolutely continuous of a continuous random variable?

I have a question in measure theory: given two measures $\nu$ and $\mu$, we say that $\nu$ is absolutely continuous of $\mu$ if for Borel set $A$ such that $\mu(A)=0$, we have $\nu(A)=0$. I want to ...
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Rigorous Bayesian Model Selection

I am learning Bayesian Model Selection. I want to understand the rigorous mathematics behind the idea of encompassing model. To be more specific, suppose we want to compare M models: Model $\mathcal{...
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Difference between tight and uniformly tight random variables?

This wikipedia page implicitly says that “tight” and “uniformly tight” random variables refers to the same concept. I find this somewhat surprising. Are there contexts in which a distinction is made ...
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understanding time-homogeneous markov chain

Could anyone help me understand the definition on page 7 definition 2.25 here? I do not understand the notation $(P(a))(A)$ - what does this mean? Also, is $P(a, A)$ a probability measure from the ...
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146 views

What is the difference between an accuracy measure and an error metric?

The two concepts are distinct in measure theory. Nonetheless, moving out from measure theory, the two terms are often used interchangeably. To most forecasters, especially forecast practitioners, they ...
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on some bounds on expectation of probability measures on discrete set

I have a situation where I have one to one correspondence between $\mathbb N$ and a subset of probability measures on $\{1,2,3,\dots, m\}$ i.e $n\mapsto p_n$, suppose for all $N, M\in \mathbb N$ $\|...
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1answer
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What does the pmf of a discrete random variable look like if it can take on the value $\infty$?

What does the pmf of a discrete random variable look like if it can take on the value $\infty$? Consider a stopping time $\tau$ of a Markov chain, which is a random variable that takes values in the ...
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Why sigma-finite measure?

I am learning measure theory and the concept $\sigma$-finite measure makes me a bit confused. Why do we need the $\sigma$-finite assumption in many important theorems?
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Sufficient condition for existence of maximal/optimal coupling

Let us consider two random variables $X$ and $X'$ defined on the same measurable space $(E, \xi)$. Is there a sufficient condition on the distributions of the random variables $X$ and $X'$ for ...
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178 views

If a probability density function (pdf) has bounded derivative, is the pdf itself bounded?

Suppose a probability density function (pdf) $f$ is differentiable almost everywhere and continuous and has a bounded derivative. Is the pdf itself bounded?
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Why we say that a probability measure $P$ is defined on $(\Omega, \mathcal{F})$?

I was wondering why we say that a probability measure $P$ is defined on $(\Omega, \mathcal{F})$, being $\Omega$ the sample space and $\mathcal{F}$ one sigma-algebra, when actually we have that $P$ is ...
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1answer
114 views

General solution of expected value of E(f(X))?

This is maybe a trivial question I came up while solving a few examples and understanding Markov/Chebyshev inequalities and subsequently in evaluating Chernoff bounds. Suppose $X$ is a random variable ...
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1answer
49 views

Defining CDFs for more than just $\mathbb R$

The usual definition of CDFs is explicitly only for random variables (as opposed to random elements), i.e., only for $\mathbb{R}$. I’ve seen definitions to extend CDFs to random vectors, i.e., $\...
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42 views

understanding recurrence relation between two measure

Hi, Could anyone explain to me how he wrote $(3.3)$ to $(3.4)$, in particular, why double integration? I am convinced with $(3.3)$ and another thing, I do not quite understand the role of $p(x,t)$. ...
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53 views

probability measure of probability measure

I if denote with $\Delta(X)$ the set of all probability measures over $X$, what is the meaning (if there is any) of $\Delta(\Delta(X))$ ?
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Integrating the inverse-Wishart density

It is alleged in this question and in the Wikipedia article and elsewhere that the density function for the inverse-Wishart distribution with $n$ degrees of freedom on $p\times p$ positive-definite ...
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1answer
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Interpreting sample space of a unknown random variable

After looking for a dozen of questions on cross validated, I've decided to write my own. We know the following mapping is called a random variable $$X:\Omega\to \mathbb{R}$$ where $\Omega$ is set of ...
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Different conditions on data measurement for ml

Is it possible to train a prediction model (on my case classifier with four classes) between data taken on different conditions? To be more specific I have two data sets and for my task I am allowed ...
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1answer
57 views

Expected value of ratio of two function of the same random variable

Let $X$ be a r.v. with absolutely continuous distribution and continuous strictly positive density $f: \mathbb{R} \rightarrow [0, \infty)$ and let $g$ a further given continuous density function. Set ...
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Uniqueness of partial covariance/corrlation if OLS is not unique

Let $X,Y,Z=(Z_1,...,Z_n)$ be random variables. Define the partial covariance between $X$ and $Y$ given $Z$ as: $$\rho_{X,Y \cdot Z} := cov(\hat{X}-X, \hat{Y}-Y)$$ where $\hat{X}$ and $ \hat{Y}$ are ...
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What does the meet of two sigma algebras mean?

I came across this notation of which I am unfamiliar; $\mathscr{F}=\mathscr{G}_{1}\vee \mathscr{G}_{1}$ where $\mathscr{G}_{1}$ and $\mathscr{G}_{2}$ are both sigma-fields of subsets of $\Omega$. It ...
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Why is a random variable being a deterministic function of another random variable mean that it is in the sigma algebra of the other variables?

Specifically, I'm learning about martingales in class right now. Given random variables $T$, $X_1, X_2, \ldots, X_n$, textbook that I'm reading draws an equivalence between the statement that the ...
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70 views

Regression with different probability measure

Say we have a set of observations $(y,X)$ where the response variable is assumed to follow some conditional distribution under a particular probability measure $Q_1$, say, $y \sim N(\mu(X),\sigma)$. ...
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1answer
148 views

Measure of stability

I am working on a machine learning project when I realized I add a question. This is not programming, nor statistic, nor a probability question, but a real pure mathematical question. So I think my ...
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1answer
19 views

Clarification regarding Markov Decision Process (MDP) formulation

Most of the reinforcement learning problems are dealt with using an MDP framework. I’m a bit confused about the formulation after reading the paper: https://arxiv.org/abs/1503.02244 In an continuous ...
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When family of DF's $\mathcal{P}$ fail to be dominated by a measure $\mu$

On the topic of minimal sufficient statistics, there is an important theorem which requires the family of probability distributions $\mathcal{P}$ is dominated by some measure $\mu$. As I understand ...
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Computation of distance between multivariate continuous empirical measures

Say we observe $X_1,...X_n$ (resp. $Y_1,...Y_n$) samples $\in \mathbb{R}^d$ drawn from a distribution $\mu$ (resp. $\nu$). Let $\hat{\mu}_n$ (resp. $\hat{\nu}_n$) be the empirical measure based on ...
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1answer
150 views

De Finetti: equivalence a.s. according to which measure

In Zen`s answer at What is so cool about de Finetti's representation theorem? that is concerned with De Finetti's 0 -1 representation theorem, he says that "De Finetti's law of large numbers" ...
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How to Bayesian update on two events which occur with measure zero?

To illustrate what I mean please consider the following hypothetical scenario: A person's favorite number $x\in[-1,1]$ is randomly distributed with atomless density function $f(x)$. Furthermore, ...
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1answer
101 views

Why does regression model theory not use measure-theoretic sigma-field type notation but counting process models do?

I have been studying counting process theory for time to recurrent event processes and am interested in the explicit use of the conditioning set in the model notation; $$E[dN(t)|\mathcal{F}_{t^{-}}]=\...
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1answer
121 views

What is the complement of an event in a sigma algebra?

Consider $\Omega=\{1,2,...,6\}$ and $F=\left\{\{1\},\{3\},\{5\}\right\}$. Let $A={1}$. If $F$ is to be a sigma algebra, then necessarily $A^c \in F$. What is $A^c$?
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“Natural” finite measure over continuous probability densities over the interval $[0,a]$ [closed]

I wonder whether there is a "natural" finite measure $\mu$ (such as the Lebesgue-Measure on $\mathbb{R}\cap[0,a]$) over the space of all continous probability density functions on $[0,a]$. EDIT: As ...
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1answer
50 views

Independence of multivariable function of random variables on the information

It might be this question was asked before, but I still want to request an answer in measure-theoretic framework. Let's define a probability space $\{\Omega, \mathcal{F}, \mathbb{P}\}$ and $\sigma$-...
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2answers
127 views

Conditional expectation given event and random variable

Let $W$ be a random variable that only takes on the values $1$ or $0$. Let $X$ and $Y$ be two other random variables. I came across the following: $$\mathbb{E}(Y|W=1, X)$$ How is this 'conditional ...
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1answer
64 views

Defining the probability space for a simple population study

I frequently work with population studies, where, let's say, age and sex are collected for $N=1,000,000$ individuals in California. I might ask a simple question: what is the average age in this ...
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330 views

Intuitive understanding of the Halmos-Savage theorem

The Halmos-Savage theorem says that for a dominated statistical model $(\Omega, \mathscr A, \mathscr P)$ a statistic $T: (\Omega, \mathscr A, \mathscr P)\to(\Omega', \mathscr A')$ is sufficient if (...