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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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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636 views

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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1answer
34 views

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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1answer
57 views

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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83 views

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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1answer
81 views

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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1answer
82 views

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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1answer
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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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3answers
293 views

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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Random likelihood function during maximizing process

Mostly common definition of Maximum Likelihood Estimator in certain parametric problem $(\mathcal{X},\mathcal{B}(\mathbb{R}^n),\mathcal{P}=\{\mathbb{P}_\theta\mid\theta\in\Theta\})$is what follows: $$ ...
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What is added by generating product sigma-algebra of two (and more) borel sigma-algebras? [duplicate]

Could someone give some example, what kind of sets delivers an extension of class $\{B_1\times B_2,\mid B_1,B_2\in\mathcal{B}(\mathbb{R})\}$ to product sigma algebra $\mathcal{B}(\mathbb{R})\otimes\...
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Different modes of stating probability law [duplicate]

I met two versions of expressing induced probability measure on $\mathbb{R}^n$ of n-dimensional random vector $\mathbf{X} = (X_1, \ldots, X_n)$ (probability law, probability distribution) defined on ...
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Identifiability vs. equivalence of probability measures

I'm a bit confused about the notion of identifiability vs. equivalence of probability measures. The following definition of identifiability I am familiar with: Let $\{P_\theta : \theta \in \...
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1answer
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Does product sigma algebra of n B(R) (borel) coincide with B(R^n)?

My question comes from different modes of probability law, which I met. I see two options in equality: set belonged to $B(\mathbb R^n)$ and cartesian product of B belonged to $B(\mathbb R)$.
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Using a single sample sequence for estimates of several integrals whose integrands have disjoint support

Let $(E,\mathcal E,\lambda)$ be a measure space $f:E\to[0,\infty)$ be $\mathcal E$-measurable with $\lambda f<\infty$ $q:E\to[0,\infty)$ be $\mathcal E$-measurable with $\lambda q=1$ and $$\{q=0\}\...
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Estimating the asymptotic variance of a specific Metropolis-Hastings estimator

Remark: I've added a detailed description of the actual setting of the application to the end of the question. I'm running the Metropolis-Hastings algorithm with target distribution $\hat\mu$ (see ...
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1answer
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Can a Metropolis-Hastings estimator converge if the proposal density vanishes whenver the target density vanishes?

I'm running the Metropolis-Hastings algorithm for a target and proposal density $p$ and $q$, respectively, for which it holds $$p(y)=0\Rightarrow q(x,y)=0\;\;\;\text{for all }y.\tag1$$ By definition ...
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Can we use of the Metropolis-Hastings importance sampling estimator here?

EDIT: Note that proposing according to $\hat Q$ is equivalent to the following scheme: Given the current state $\tilde x=(i,x')\in\tilde E:=I\times E'$ and $x:=\varphi_i(x')$, sample $j\sim\tilde T(\...
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1answer
219 views

Does Bayes theorem apply to joint distributions of discrete and continuous random variables?

Bayes theorem is defined for both discrete variables in terms of probabilities and continuous variables in terms of densities. If random variables $X,Z$ are jointly distributed, with $f_X(x)$ ...
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1answer
48 views

Number of observations to study the reproducibility

I'm running an experiment which is about to investigate the influence of gases on the resistance of sensors. Since this is chemistry and gases are experimentally rather hard to handle, I would like to ...
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278 views

Do likelihood functions require sigma-algebras as probability spaces do?

In statistics, the likelihood function (often simply called the likelihood) is formed from the joint probability of a sample of data given a set of model parameter values; it is viewed and used as a ...
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1answer
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a question about a proof here involving real analysis or measure theory

i have a question about a proof here that I was reading: Basically what I don't understand is the last sentence of the proof where it says: $Pr{\{t<S_{n+1} \leq t + \delta\}} = f_{S_{n+1}}(t)(\...
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A Measure Theoretic Formulation of Bayes' Theorem

I am trying to find a measure theoretic formulation of Bayes' theorem, when used in statistical inference, Bayes' theorem is usually defined as: $$p\left(\theta|x\right) = \frac{p\left(x|\theta\right)...
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55 views

Can we sample from the wrapped normal distribution and evaluate the density of the sample simultaneously?

In a computer program (written in C++), given $x\in[0,1)$ and $\sigma>0$, I need to sample $y$ from the wrapped normal distribution $\mathcal W_{x,\:\sigma^2}$ with mean $x$ and variance $\sigma^2$ ...
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1answer
89 views

Transforming any probability density over $R^N$ to any other probability density

If $p_1(x)$ and $p_2(x)$ are two arbitrary probability density functions defined over $R^N$, i.e. both are non-negative and properly normalized so that $$ \int d^N x \, p_1(x) = 1 = \int d^N x \, p_2(...
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1answer
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question about meaning of indistinguishable in an example in stochastic process

I have a question about example 3.4 on Page 6 of the document here: http://stat.math.uregina.ca/~kozdron/Teaching/Regina/862Winter06/Handouts/revised_lecture1.pdf My specific question is that I ...
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1answer
48 views

Unbiased Metropolis-Hastings estimator of the form $\frac{\sum_{i=1}^nW_if(Y_i)}{\sum_{i=1}^nW_i}$. How do we need to choose $W_i$?

Let $(X_n)_{n\in\mathbb N_0}$ be the Markov chain generated by the Metorpolis-Hastings algorithm with proposal kernel $Q$ and target distribution $\mu$ and $(Y_n)_{n\in\mathbb N}$ denote the ...
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42 views

Minimizer of $\int\mu({\rm d}x)\int\kappa(x,{\rm d}y)|g(x)-g(y)|^2$ for a jump kernel $\kappa$ of the Metropolis-Hastings algorithm

Let $\kappa$ be a sub-Markov kernel on a measurable space $(E,\mathcal E)$ and $\mu$ be a probability measure on $(E,\mathcal E)$ reversible with respect to $\kappa$. Assume $\kappa$ and $\mu$ admit a ...
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1answer
132 views

“Independent observations” via measure theory

I'm reading Chernoff's paper "A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations," and am trying to understand it in terms of measure theory. On page 495, it ...
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1answer
134 views

Where is the measure theoretic probability theory actually applied?

Where is measure theoretic probability theory actually applied? I've done quite a bit of graduate work in machine learning, Bayesian machine learning, information theory, and statistics (both ...