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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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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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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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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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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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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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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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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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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23 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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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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291 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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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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1answer
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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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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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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165 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
80 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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1answer
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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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1answer
62 views

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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38 views

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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2answers
60 views

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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1answer
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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
57 views

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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28 views

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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1answer
280 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
140 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
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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., $\...