# 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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### 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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### 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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### 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.
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### 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 ...
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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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### 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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### 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 ...
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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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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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-...
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### 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, <...
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### 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 ...
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### 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 ...
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### 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....
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### 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?
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### 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: ...
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 ...
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 ...