Questions tagged [exchangeability]

A set of random variables is exchangeable when their joint distribution is invariant under any permutation of the random variables.

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What GEE-exchangeable method can do that robust variance can't?

I asked a related question before here on the difference between GEE method with exchangeable varcov structure v. Robust standard errors known as Huber White method in group randomized trials. As ...
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137 views

Exchangeable Processes over the Simplex

You are likely all familiar with Polya Urn process. I initially start with an urn containing $b$ black balls and $w$ white balls. At each step, I sample a black ball with probability $\frac{b}{b+w}$ ...
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116 views

Correlated Bernoulli Trials

Suppose there are $n$ dependent Bernoulli trials, $X_{1}$,...,$X_{n}$ with $% X_{j}\in \{1,0\}$ and $\Pr (X_{j}=1)=q$ for all $j=1,...,n$. For any $% n\geqslant 2$ dependent Bernoulli trials, in the ...
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81 views

Deeper proof of common expected value of random convex weights

$\newcommand{\E}{\mathbb{E}}$Let a finite collection of exchangeable random variables $X_1,...X_n$ (some authors would call this collection "interchangeable" since it is finite, reserving "...
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91 views

From bivariate to trivariate probability distribution

Let $\mathcal{G}$ be the space of all possible bivariate probability distributions. Let's pick a bivariate probability distribution $g\in \mathcal{G}$. Can we always find a random vector $(X,Y,Z)$ ...
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1answer
182 views

When are observations not weakly exchangeable?

In the book "Common errors in statistics" https://www.amazon.com/Common-Errors-Statistics-Avoid-Them/dp/1118294394, I read the following statement Permutation tests only yield exact significance ...
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78 views

Are random variables sampled upon stopping rules exchangeable?

In this article from D. Berry https://www.jstor.org/stable/2684222?seq=1#page_scan_tab_contents the author uses an example to introduce some limits of p-values in frequentists analyses. He takes an ...
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Intuitive understanding of the Aldous-Hoover representation theorem for row-column exchangeable arrays

I would like to ask a couple of questions about the Aldous-Hoover theorem for the representation of probability distributions over (2D) arrays with exchangeable rows and columns. I'd be happy about ...
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93 views

What does it mean “a conditional expectation given a stochastic process”?

Let $(\Omega,\mathscr{F},P)$ be a probability space. Let $X,Y$ be random variables on $\Omega$. Then, we say $Z\sim X|Y$ iff (i) $\int_{Y^{-1}(A)} X dP = \int_{Y^{-1}(A)} Z dP$ and (ii) $Z$ is $\...
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Can I predict if my sequence is exchangeable?

Supppose I have a sequence as so (which we can really think of as a sequence of smaller sequences): [[1 1 1 1 1 1], [2 2 2 2 2 2], [3 3 3 3 3 3], [4 4 4 4 4 4]] ...
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1answer
61 views

When $(X_1-X_0, X_1-X_2)\sim (X_2-X_0, X_2-X_1)\sim(X_0-X_1, X_0-X_2)$?

Consider a bivariate probability distribution $P: \mathbb{R}^2\rightarrow [0,1]$. I have the following question: Are there necessary and sufficient conditions on the CDF associated with $P$ (joint or ...
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How is exchangeability related to covariate shift?

I understand that exchangeability refers to the notion that the order of data in a sequence does not affect the joint distribution of that data. In a sense, the current data we possess is from the ...
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143 views

Expected value of maxima of dependent random variables

I don't know if such theorem exists, but what I am looking for is a closed-form solution for $$E[\max(X_1, ..., X_N)]$$ where $X_1, ..., X_N$ is a sequence of dependent identically distributed ...
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1answer
155 views

Sharing of information and borrowing of strength. What models are there

I am currently working on identifying all the methods that have been implemented in all kinds of disciplines in order to borrow strength. If you have ever used, or came across such methods i would be ...
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65 views

When is an ellipsoidally (elliptical) distributed random variable spherically symmetric?

If $Y$ is ellipsoidally distributed, and $\mu_y \propto 1_p$ and $\sum_y = \sigma^2 I$ is $Y$ spherically symmetric? EDIT: added - Is it exchangeably distributed? Please give a proof not a yes/no.
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Can we use permutation test everywhere to check difference when the null hypothesis is “there is no difference between two groups of data?”

What I am trying to do? Let's assume, I am working on a research problem where only one participant is present. I have a data set consisting that participant's respiration rate (The number of breaths ...
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142 views

How to regularize when facing exchangeable vectors in deep learning?

I am seeking to estimate a regularization method to estimate the conditional probability $p(y|\mathbf{X})$ where $\mathbf{X}\in\mathbb{R^{n\times m}}$ and where the probabilities are invariant by ...
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154 views

How the De Finetti's Representation Theorem works in this case?

For the special case of infinite sequence of $\{0,1\}$ valued random variables the theorem is stated as $$ Pr(x_1, \ldots, x_n) = \int_0^1 p^{(\sum_{i=1}^n x_i)}[1-p]^{(n - \sum_{i=1}^n x_i)} dQ(p)\,. ...