# 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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### Is the converse of de Finetti's theorem true?

As I understand it from Wikipedia, de Finetti's theorem says: "Exchangeability implies that variables are conditionally independent given some latent variable". Is the converse true as well?
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### Multivariate Hypergeometric distribution for non-exchangeable sequence

I have a problem that I think is similar to the Multivariate Hypergeometric distribution urn problem as described on Wikipedia. I have a dataset where each row is a trial. All trials have the same ...
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### Show that the variance of the longitudinal estimate is $\dfrac{2 \sigma^2(1-\rho)} { n}$ rather than $\dfrac{\sigma^2(\rho)}{n}$

patients are randomızed to elther treatment, their pressures are measured at baseline, treatment is administered for two weeks and $\mathrm{BP}$ is then measured a second time. The treatment effect is ... 28 views

### Are there examples of ML or stats approaches that are valid for IID data, but not exchangeable data?

Lots of supervised learning theory is motivated using the IID assumption. Do most of these methods apply equally well if data is only exchangeable, and not IID? Can you provide an example where this ...
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### How to derive g-computation for a longitudinal experiment from sequential conditional exchangeability?

We have a longitudinal experiment, with interventions $\bar{A}=\{A_1,A_2,\ldots,A_K\}$ and outcomes $\bar{Y}=\{Y_1,Y_2,\ldots,Y_K\}$. Using some sequential conditional exchangeability assumptions, it ...
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### Switching interventions after randomization

Suppose in a randomized control study, we assigned each individual to one of two interventions (A or B) with equal probability. And suppose for the sake of argument, there is complete balance in ...
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### How do you calculate the expected value of a discrete distribution without replacement?

Say I have a set of 10 values I want to draw 3 values from, uniformly, without replacement. For instance: $$S = \{0,0,0,0,22.95,0,0,0,19.125,25.5\}$$ With replacement, this seems simple, you just add ...
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### Exchangeability, causal inference, and partial pooling

In Statistical Rethinking, Richard McElreath writes the following concerning the use of partial pooling (i.e. varying/random effects) in Bayesian hierarchical models: Could we also use partial ...
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### Proof for a seemingly simple property for fully-connected coloured random graphs?

I have a probability distribution defined over a set of fully-connected simple graphs depending on their coloring. Let us have a fully connected graph with $N$ nodes, a node may have a color $i$ ...
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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 $X_{(... • 9,641 2 votes 0 answers 59 views ### How is exchangeability related to covariate shift? I understand that exchangeability refers to the notion that the order of data in a sequence does not affect the joint distribution of that data. In a sense, the current data we possess is from the ... 15 votes 4 answers 647 views ### If$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$. Attempt: Please check if the below is correct. Let ... • 283 3 votes 0 answers 103 views ### From bivariate to trivariate probability distribution Let$\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)$... • 538 8 votes 2 answers 252 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 ... • 2,781 2 votes 1 answer 764 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 ... 3 votes 1 answer 133 views ### Can every parameter$\Theta$in Bayesian modelling be explained via De Finettis representation theorem My question is the following: I recently got to know (and love) De Finettis representation theorem and I now started to read a Book an Bayesian statistics. However this book simply takes as the ... • 2,781 5 votes 1 answer 406 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 ... • 486 2 votes 1 answer 1k 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 ... • 165 1 vote 1 answer 266 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 ... • 1,017 3 votes 1 answer 151 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 ... • 107k 11 votes 3 answers 4k 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 ... • 321 2 votes 1 answer 444 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 ... • 321 3 votes 1 answer 86 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 ... • 486 3 votes 1 answer 50 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 ... • 2,781 7 votes 2 answers 537 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 ... • 2,365 2 votes 1 answer 64 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-... • 924 1 vote 1 answer 304 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)})_{...
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 ...