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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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family of probability measure, sub probability measure [closed]

For a Polish space $X$, what does it mean by a family of measures $\{M_x: x\in X\}$? Could anyone give me an example? Also, what is the family of sub-probability measure $\{P_x:x\in X\}$? Thanks for ...
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45 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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1answer
80 views

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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63 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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Difference between several/Multivariate Random Variables (RVs) & a Sequence of RVs?

What is the difference between several/Multivariate Random Variables (RVs) & a Sequence of RVs? Example: Picking a student from class and noting his/her height and weight is several RVs (...
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1answer
134 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
16 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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1answer
41 views

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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76 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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1answer
200 views

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
84 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
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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
34 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
86 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
49 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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173 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 (...
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1answer
36 views

How do I create an iid Rademacher sequence?

The lecture notes say: Let $(\Omega,\mathcal{A},P) = ((0,1],\mathcal{B}((0,1]),\lambda)$ where $\lambda$ is the Lebesgue measure on the unit interval. Define $X(\omega) = 1$ for $\omega > 1/2$ ...
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1answer
88 views

When does the Wasserstein metric attain inequality WLOG?

I’m reading a classic paper [1] that describes a version of the Wasserstein metric (aka Mallows metric), defined as follows. Let $F$ and $G$ be probabilities in $\mathbb{R} ^B$, and let $U \sim F$ and ...
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Sufficient conditions for absolute continuity of a distribution function

Consider two random variables $X,Z$ defined on a probability space $(\Omega, \mathcal{F}, P)$. Assume 1) Let $\mathcal{Z}$ denote the support of $Z$. For any $z\in \mathcal{Z}$, the support of $X$ ...
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dependency of the deviation to the measurement level: how to test normality?

A machine measures the height of some plants. In the context of industrial quality procedures, we repeat the same measurement Hi 10 times and expect the ...
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174 views

Why can't the complete class theorem be easily generalized to all locally-compact spaces?

So I was reading Christian P. Robert's The Bayesian Choice, going through the constellation of results related to complete class theorems, and I don't see why all of them are necessary. In particular, ...
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Is probability theory the study of non-negative functions that integrate/sum to one?

This is probably a silly question, but is probability theory the study of functions that integrate/sum to one? EDIT. I forgot non-negativity. So is probability theory the study of non-negative ...
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1answer
351 views

Is convergence in probability equivalent to “almost surely… something”

Convergence in probability is weaker than almost sure convergence. I wonder if the property "$X_n$ converges to $0$ in probability" can be expressed as: "For almost all $\omega$, the sequence $X_n(\...
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1answer
66 views

Clarification of MCMC failure example (Roberts and Rosenthal, 2004)

I am reading the 2004 MCMC paper by Roberts and Rosenthal and was hoping someone could shed some further light on Example 3. I restate Theorem 4 first, which the example relies on. Theorem 4. If a ...
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1answer
59 views

Why is the Dirichlet Process not a completely random measure?

A completely random measure assigns independent mass to nonintersecting subsets. I cannot figure, however, how the Dirichlet Process does not qualify as a CRM? Aren't the atoms all independent from ...
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1answer
149 views

How to justify a KL divergence when a distribution contains continuous and discrete components

Like in contamination models, some distributions have discrete component. e.g. $p(x) := (1 - \epsilon) q(x) + \epsilon \delta_{x_0}(x)$ In these distributions, is there a way to justify a definition ...
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1answer
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Bivariate normality and statistical work when the joint density is not defined

Let $X$ follow a Normal distribution, and let $Y = a+bX$ (not both constants zero). The normal distribution is closed under shifting and scaling, so $Y$ also follows a normal distribution. If I ...
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1answer
745 views

Interpretation of Radon-Nikodym derivative between probability measures?

I have seen at some points the use of the Radon-Nikodym derivative of one probability measure with respect to another, most notably in the Kullback-Leibler divergence, where it is the derivative of ...
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Complete randomness imply Poisson process

In this paper, the authors claim that Theorem 1. A random point process $\Pi$ on a regular measure space is a Poisson process if and only if $N_\Pi$ defined by $N_\Pi(A) = \#\{\Pi \cap A\}$ is a ...
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Measure-theoretic derivation of change of variables formula for probability density functions?

Assume we have a $2$-dimensional sample space $(\Omega, B, P_\Omega)$, with $\Omega =\mathbb R^2$ with borel measure and probability measure $P$, where the axes are simply equal to random variables $...
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1answer
49 views

custom value transformation function [closed]

Background: Blast is a (famous) tool for finding high scoring local alignments between sequences. It allows you to set a search parameter that controls the statistical significance of each (similar) ...
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1answer
85 views

Can we define for a probability measure, a random variable to be its “density function”?

I am trying to relate the conditional probability (a random variable) $P(A|G)$ to the probability measure $P(A)$ ( a measure). It is weird to me that while the probability $P$ is a measure, the ...
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67 views

Textbook or Video Course in probability [duplicate]

I am an engineering student and I studied probability textbook by Sheldon Ross and Engineering Statistic by Montgomery. However, I need to know more about the mathematics of probability and statistics....
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How to recover slice-curve $g$ from a multivariate probability distribution $F$ on $\mathbb{R}^n$

Let $P$ be a probability measure on $\mathbb{R}^n$ and $F$ be the corresponding distribution function. Also let $f:\mathbb{R}^n \to \mathbb{R}$ be the 'default' density function of $F$ (i.e. the radon ...
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1answer
98 views

Prove that MLE does not depend on the dominating measure

The question is already posted here : https://math.stackexchange.com/questions/2530202/prove-that-mle-does-not-depend-on-the-dominating-measure Let $(X,\mathbb{F})$ be a measurable space, $\left\{P_{\...
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0answers
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definition of circular/directional random variable

Good day, i wanted to ask you to check my definition of circular random variable since i have never seen any. So based on the assumption, that difference between linear and circular variables, is ...
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0answers
46 views

Absolute continuity of distribution function

I have a question about the distribution function $F(x)$. There is statement in (K.V.Mardia, Statistics of directional data) about the circular distribution function (which has all the properties of ...
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1answer
51 views

Why must a random variable be $\mathcal{F}$-measurable?

Needless to say, I know that the answer is trivially "because that's part of the definition of a random variable", but what I'm really looking for is why that's part of the definition. Why do we want ...
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63 views

Where can I find a measure-theoretic statement of the Von Neumann-Morgenstern axioms?

Most of the discussion of the Von Neumann-Morgenstern axioms I have seen isn't fully formal - in fact, a lot of it only applies to finite probability spaces. Where can I find a more general discussion,...
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1answer
45 views

Proving this statement without the independence condition (Borel 0-1 Law)

I am trying to prove this statement, "if $P(S_n) \to 1$ as $n \to \infty$, prove that there exists subsequence $\{n_k\}$ such that $P(\cap_{n_k}S_{n_k}) > 0$". As $lim_{n \to \infty} P(S_n) = 1 \...
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3answers
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What do the elements of a sample space in a probability space represent in a real-world statistical test?

Consider the measurable space $(\Omega,\mathcal{F})$ where $\Omega$ is the sample space and $\mathcal{F}$ is the a set of observable events. Further assume that we are estimating a linear regression $...
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1answer
42 views

Find $\sigma$-field $\sigma(\mathscr A)$ generated by $\mathscr A$

Let $\Omega =\{1,2,3,4\}$, and let $\mathscr A = \{\{1\},\{2\}\}$. Find $\sigma$-field $\sigma(\mathscr A)$ generated by $\mathscr A$. My answer is $\sigma(\mathscr A) = \mathscr A \cup \mathscr A^c =...
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1answer
167 views

Plotting the typical set of a Gaussian distribution

There is this article where the author Michael Betancourt uses this image to convey the concept of the typical set in a distribution. I would like to plot the typical set of a univariate or a ...
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0answers
59 views

Interpreting base measure in exponential family as an improper prior (because entropy)

Long-time listener, first-time caller. I'm reading the Wikipedia pages on exponential families and maximum entropy probability distributions, and trying to wrap my head round the role of the base ...
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42 views

What does it mean to think of a population as a probability measure?

I am working independently through parts of Jun Shao's Mathematical Statistics after spending a year on Casella and Berger. Most of the measure theory related content is pretty new to me. In the ...
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1answer
1k views

Why IQ test “discrete scores” are said to have “Normal shape”

IQ test scores don't increment in decimals, but in whole numbers (e.g., 138, 140, 150 etc.). In other words, IQ tests only provide discrete scores [not continuous] (No body can get an IQ score of $115....