Questions tagged [exchangeability]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
15 views

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 ...
3
votes
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 ...
-1
votes
1answer
20 views

Is it possible to have dependent and exchangeable random variables? [duplicate]

Is it possible to have dependent and exchangeable random variables? Someone told me yes.
4
votes
5answers
201 views

Is independence subjective?

I am trying to better understand exchangeability. Suppose that I would like to run an experiment. I will pick two people and give each a fair coin and tell them to toss N times in a row. The ...
8
votes
0answers
768 views

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 ...
1
vote
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 ...
3
votes
3answers
262 views

Relationship between exchangeability and independence

If the random variables $X_1,...,X_n$ are I.I.d. when conditioned on an unknown parameter $\theta$, where $p(\theta)$ represents our prior beliefs about $\theta$, it follows from the commutativity of ...
1
vote
0answers
47 views

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 ...
1
vote
1answer
91 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 $\...
6
votes
1answer
119 views

Correlation between an observation and its rank in a random sample

Suppose $X_1,X_2,\ldots,X_n$ are i.i.d random variables with an absolutely continuous distribution. We say the observation $X_i$ has rank $R_i$ if $$X_i=X_{(R_i)}\quad,\,i=1,2,\ldots,n,$$ where $X_{(...
5
votes
2answers
491 views

GEE model returns GLM results

I have simulated some longitudinal data with 100 subjects and 7 measurements per subject. My data has random intercept which will induce "exchangeable" correlation matrix. My goal is to fit two ...
3
votes
2answers
127 views

Permutation test comparing nested non-linear models with an exchangeable dummy variable

This question is closely related to an earlier question, but I realized my case was actually a lot more specific than the way I formulated it there, in ways that I think merit a separate answer. I ...
2
votes
0answers
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 ...
4
votes
0answers
115 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 ...
5
votes
1answer
1k views

Fixed Regressor Conspiracy and Connection to Exchangeability

In simple regression model regressors are treated as fixed rather than stochastic. Whoever picks the experimental values for the regressors, decides in which frequency to include each value. This can ...
1
vote
1answer
164 views

Sampling from a finite sequence without replacement yields exchangeable sequences?

I have read (and re-read) the wikipedia article on Exchangeability https://en.wikipedia.org/wiki/Exchangeable_random_variables . The disconnect for me is that : after having sampled without ...
1
vote
1answer
126 views

Understanding exchangeability

Suppose you had information of a true success rate of passing an exam from different schools, and that the 'true' success rates are similar, can you assume exchangeability is applicable to these ...
1
vote
1answer
110 views

Exchangeability of group effects

Regarding this basic model with an individual-level covariate $x_{ij}$, group effects $U_{0j}$, and individual effects $R_{ij}$: $Y_{ij}=\gamma_{00}+\gamma_{10}x_{ij}+U_{0j}+R_{ij}$ the multilevel ...
1
vote
0answers
41 views

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 ...
0
votes
0answers
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)\,. ...
1
vote
0answers
142 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 ...
2
votes
1answer
53 views

Can every parameter $\Theta$ in Bayesian modelling be explained via De Finetti`s representation theorem

My question is the following: I recently got to know (and love) De Finetti`s representation theorem and I now started to read a Book an Bayesian statistics. However this book simply takes as the ...
1
vote
1answer
71 views

Covariance of an exchangeable random vector

I am trying to solve the next exercise but I don´t know how to start. Let $X\in\mathbb{R}^n$ be a random vector which is exchangeable in the sense that $X$ has the same distribution as $(X_{\pi(i)})_{...
6
votes
0answers
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 $\frac{b}{b+w}$ ...
4
votes
1answer
151 views

Wilcoxon signed rank test independence assumption

Let us say we perform Wilcoxon signed-rank test on paired samples $x_{1,i}$ and $x_{2,i}$. I am trying to understand the independence assumption of the test. My questions are: Which quantities must ...
5
votes
1answer
5k views

If random variables are drawn from an identical distribution, why doesn't this guarantee they are independent?

Having read a little about exchangeability, I went back to thinking about the iid condition required for the central limit theorem. It struck me that if two random variables are drawn from an ...
3
votes
2answers
218 views

Are we ignoring implications by de Finetti's theorem on regression?

De Finetti's theorem states that, if observations $(x_1, x_2, x_3, \cdots)$ are infinitely exchangeable, then their joint probability $p(x_1, x_2, \cdots, x_N)$ has a representation as a mixture: $$p(...
2
votes
1answer
406 views

When are time series data exchangeable?

E.g. is a linear process exchangeable? What about strict vs weak stationarity? EDIT: for clarity, I'm asking if/when the sequence of data points $(x_1, x_2, \ldots, x_n)$ in the time series is ...
7
votes
2answers
259 views

In a Bayesian hierarchical model, if exchangeability doesn't hold, what exactly goes wrong?

In many textbooks, when a Bayesian model is presented, such as a classic Normal-Normal model, there is some sort of brief mention that the trials must be exchangeable. I am wondering why this is ...
1
vote
1answer
74 views

Simple question on exchangeability

We say that $(x_1,x_2,\dots)$is an infinitely exchangeable sequence of random variables iff for any permutation $\pi$, $p(x_1,\dots,x_n)=p(x_{\pi(1)},\dots,x_{\pi(n)})$. Let $(x_1,x_2,\dots)$ be an ...
4
votes
3answers
565 views

Bayesian models and exchangeability

I did not see that explicitly mentioned, even though I think it is correct. Isn't the exchangeability assumption the most common assumption about examples in the Bayesian setting? I am thinking of a ...
3
votes
1answer
64 views

Exchangeability and data smoothing

If a non-i.i.d sequence of a continuous random variable that is exchangeable, is smoothed by taking rolling average, is the resulting sequence exchangeable? My intuition suggests that it is not. I'll ...
1
vote
0answers
35 views

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]] ...
1
vote
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 ...
4
votes
0answers
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 "...
1
vote
0answers
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.
3
votes
0answers
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 $(X,Y,Z)$ ...
5
votes
2answers
916 views

Exchangeability and IID random variables

Every IID sequence of random variables is considered to be exchangeable, i understand why its necessary for the random variables to be identically distributed to assume exchangeability, but why the ...
8
votes
2answers
147 views

Are products of exchangeable RVs exchangeable?

Assume that $$X=(X_1, ..., X_n),: (\Omega, A,P)\to (\{0,1\}^n, 2^{{\{0,1\}}^n})$$ and $$Y=(Y_1, ..., Y_n):(\Omega, A,P)\to (\{0,1\}^n, 2^{{\{0,1\}}^n})$$are two random Variables that have binary RVs ...
3
votes
1answer
42 views

What role does stochastic independence have for Bayesians

I recently read a chapter about Subjectivist notions of Probability in Gillies (2012) and I stumbled upon the concept of exchangeability as the Bayesian version of independence (or something similar ...
3
votes
1answer
117 views

Create an exchangeable sequence from a non-exchangeable sequence

Suppose you have an arbitrary sequence of real values $\{ a_i | i \in \mathbb{N} \}$. Now, suppose you want to randomise the order of this sequence so that it is now exchangeable. To do this, you ...
2
votes
1answer
50 views

Latent Modeling of Non-exchangable data or de Finneti on time-series

De Finneti's theorem applies on infinetely exchangeable sequences and allows building probabilistic models with latent variables that allow us to describe them. Is there a similar theorem for time-...
16
votes
1answer
2k views

What is the intuition behind exchangeable samples under the null hypothesis?

Permutation tests (also called a randomization test, re-randomization test, or an exact test) are very useful and come in handy when the assumption of normal distribution required by for instance, <...
4
votes
2answers
391 views

What are some statistical tests for exchangeability of a data set?

The representation theorem of de Finetti is seen by some as motivation for the use of Bayesian and/or hierarchical modeling. In some settings, it may be plausible to assume measurements are ...
41
votes
1answer
14k views

Can someone explain the concept of 'exchangeability'?

I see the concept of 'exchangeability' being used in different contexts (e.g., bayesian models) but I have never understood the term very well. What does this concept mean? Under what circumstances ...
5
votes
2answers
591 views

Partial exchangeability - Theory and results

De Finetti has defined in 1938 the concept of partial exchangeability as a weaker form of exchangeability in order to deal with situations in which the symmetry among all observations is not desirable....
9
votes
3answers
1k views

Why the exchangeability of random variables is essential in hierarchical bayesian models?

Why the exchangeability of random variables is essential for the hierarchical Bayesian modeling?
5
votes
1answer
447 views

How to test the exchangeability of data?

I have implemented a constraint-based random number generator producing 3 columns with the constraint that each row sums to 1: ...
6
votes
2answers
947 views

intuition behind exchangeability property, and its use in statistical inference

I'm reading "Bayesian Data Analysis" by Gelman et al., and I encountered this exchangeability property: $\{X_n\}_{n \in N}$ is exchangeable if $F_{X_1,\ldots,X_n}(x_1,\ldots,x_n)$ is symmetric in its ...
0
votes
0answers
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 ...