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
49
votes
1answer
16k 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
1answer
6k 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 ...
7
votes
2answers
2k 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 ...
15
votes
4answers
391 views

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 ...
18
votes
1answer
3k 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, <...
1
vote
1answer
97 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 ...
7
votes
3answers
382 views

Permutation tests and exchangeability [duplicate]

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 ...
4
votes
2answers
581 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 ...
4
votes
1answer
200 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
2answers
713 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....
1
vote
1answer
148 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 ...
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?
6
votes
2answers
1k 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 ...
4
votes
2answers
251 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(...
6
votes
5answers
257 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 ...
3
votes
2answers
173 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
1answer
215 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 ...
6
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 ...
5
votes
1answer
477 views

Why are words in a document for bag-of-words model exchangeable but not independent?

I've been watching a talk (section between 07:20-08:00) given by Michael Jordan and I'm getting confused between independence and exchangeability. He says that "If we have a document and you ...
1
vote
2answers
1k views

Which probability distributions are not exchangeable?

Are there any specific families of probability distribution which are not exchangeable by construction? I was thinking that the Hyper Geometric distribution would not be since it models random ...
1
vote
1answer
349 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 ...