Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [exchangeability]

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

Wikipedia has an article https://en.wikipedia.org/wiki/Exchangeable_random_variables with further references.

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What is so cool about de Finetti's representation theorem?

From Theory of Statistics by Mark J. Schervish (page 12): Although DeFinetti's representation theorem 1.49 is central to motivating parametric models, it is not actually used in their implementation. How is the theorem central to parametric…
gui11aume
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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 is this concept invoked and why?
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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, t-test is not met and when transformation of the…
doctorate
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If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$

Question If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$. Attempt: Please check if the below is correct. Let say, we take the sum of the those conditional…
learning
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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 need for independence, (or is there a need)? In the…
s.g
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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?
user3125
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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 Macro pointed out Freedman in his 2006 paper The…
Sam
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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 identical distribution, the occurrence of one event does…
HFC
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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 as their components (Therefore…
Sebastian
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Difference between exchangeability and independence in causal inference

When inferring causal effects from observational studies, one of the assumptions that's generally required is the exchangeability assumption. Suppose $A \in \{0, 1\}$ is a binary treatment, and let $Y^a$ denote the counterfactual outcome under…
Devon
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Permutation tests and exchangeability

Permutation tests assume exchangeability of the response/observations under the null hypothesis. In what practical situations is this clearly violated? When is it unproblematic? Edit/additional question in order to not be considered as duplicate: if…
Michael M
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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 $\frac{b}{b+w}$ (alternatively, a white ball with probability…
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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 necessary and what goes wrong if exchangeability…
user321627
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How to find the prior distribution in De Finetti Representation Theorem?

I am working with a Polya urn made of red ($r$) and white ($w$) balls. For each extracted ball, I put it back in the urn together with $c=1$ balls of the same color. I have computed the following joint probability law of $n$ variable $Y_i$ for…
PhDing
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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 arguments $\forall n \in N$. I understand the…
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