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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Measureability of preimage of generating class is enough- proof

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, and $X(\cdot):\Omega\rightarrow \mathbb{R}$ be a random variable. We know, that $X^{-1}[\mathcal{B}(\mathbb{R})]=\{S\in\mathcal{F}\mid X^{-...
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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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49 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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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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79 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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78 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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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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43 views

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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42 views

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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62 views

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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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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103 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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47 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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262 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
102 views

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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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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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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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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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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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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“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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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 ...
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26 views

Identifiability of a probability given a set of conditional independence statements and distributions

I am seeking help for finding papers demonstrating the identifiability of a probability given a set of conditional independence statements and a set of probability distributions. More specifically, I ...
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64 views

Help to understand martingale example from Billingsley

I am studying Billingsley's section 35 about martingales and I have some difficulties understanding one of the examples. This is the example 35.4. Suppose we have a measurable space $(\Omega,\mathcal ...
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Minimizing the rate of geometric ergodicity of a Metropolis-Hastings kernel

Let $\kappa$ denote the transition kernel of the symmetric Metropolis-Hastings algorithm with proposal kernel $Q$ and target distribution $\mu$. What would be a general guideline if we're aiming to ...
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26 views

almost sure convergence concepts applying to possibly non random variable

I have a question about almost sure convergence of something. Basically it is coming from confidence interval. I was reading that $\liminf\limits_{n\rightarrow\infty} \frac{1}{n} \sum_{i=1}^{n} I_{\{...
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Is [0,1] x $y$ a measurable set in the Borel algebra?

Let's say I have a joint distribution $p(x,y)$ where $x \in [0,1]$ and $y \in \mathbb{R}$. I'm wondering if $[0,1] \times y$ is measurable in this case. It seems like it shouldn't be, but I came ...
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222 views

Distribution over functions that integrate to 0

This question is about Gaussian processes interpreted as distributions over the space of functions. Gaussian processes have the property that their integrals are Gaussian random variables; cf. this ...
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Laws of large numbers for sample covariances when each random sample are just known to be integrable random vectors

Cross posted here: https://math.stackexchange.com/questions/3407149/laws-of-large-numbers-for-sample-covariances-when-each-random-sample-are-just-kn Let $X_1,...X_n$ be $p$-dimensional random sample,...
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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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323 views

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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26 views

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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67 views

$P(h\leq k)=P(t \leq k),\forall k\in \mathbb{R}$ 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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53 views

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=...