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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What does 'Exchangeability' mean in multiple testing?

In A practical guide to methods controlling false discoveries in computational biology, the author mentioned 'Exchangeability': According to Wiki, the definition of 'Exchangeability' is : ...
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How useful is exchangeability for non-extendible sequences?

Question How to model the predictive distribution $\mathrm{P}(X'\mid X_1=x_1, \dotsc, X_n=x_n)$ if I know that the sequence $X_1, \dotsc, X_n, X'$ is exchangeable, but I cannot assume it to be $N$-...
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Derivation of g-formula

I am looking over a paper that derives the g-formula. The setting is the following: two treatments $A_0, A_1$ and a sole covariate $Z$ The assumptions are: Counterfactual consistency: $E[Y|A_0=a_0,...
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Can permutation tests only prove non i.i.d-ness?

I just learned about permutation tests through self-study. According to Wikipedia we have : (1) "The null hypothesis is that all samples come from the same distribution $H_{0}:F=G$." and : (...
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Is a random subset of exchangable data still exchangable? How about iid?

I am dividing the data that I work with (iid, thus also exchangable) into training and calibration sets based on the random sampling (drawn from Bernoulli dist with parameter q). Does this still ...
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Fundamental theorem of card counting: exchangeability and conditional distributions

I have some elementary queries about the relationship between exchangeability and conditional distributions. For theorem 11.6.4 in the Doctrine of Chances, Ethier S. (2010) it is proved that $\{Z_n\}_{...
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exchangeability in LDA

Can someone help me understand how exchangeability works for Latent Dirichlet Allocation (LDA) and how it enables us to treat words in the test set that are not present in the training set? I know ...
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procrustes alternative

Im comparing two multidimensional MDS solutions, the solutions have the same number of dimensions. I don't think I can use the permutation version of procrustes analysis (commonly, PROTEST in R::vegan)...
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Is this a valid statement according to the de Finetti theorem?

Some background on my problem - Let us consider a discrete memoryless channel (DMC) $W_{Y|X}$ from Alice to Bob. A DMC is a conditional probability distribution over the random variable $Y$ given ...
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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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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 ...
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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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Proving Independence due to exchangeability?

I have a set of bernoulli random variables $\{x_i\}^{n}_{i=1}$ and $\{x_{ij}\}_{i< j}$. They have a probability distribution with following conditional independence: $$P(\{x_i\}^{n}_{i=1},\{x_{ij}\}...
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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^...
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Conditional exchangeability when conditioning on continuous variable

Conditional exchangeability is often introduced in a simple setting with a binary outcome, binary treatment, and binary confounding variable. In this setting, exchangeability holds within strata of ...
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Practical example for (conditional) ignorability in potential outcomes

Within the potential outcomes framework (Rubin Causal Model), I find that most sources efficiently explain ignorability and conditional ignorability algebraically. As far as I can tell, some use "...
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correcting t-test for dependent but not paired data (in R)

My data $\{X_1,...X_N \}$ are assumed exchangeable with known correlation $\rho$. I want a "t-test" (or similar) of mean $E(\bar{X})=0$ in the R language. More generally, i seek an easy and ...
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Expectation of uniform variates

Let $X_{1},X_{2},X_{3}$ be random variates from $U(0,1)$. It is required to compute $E(\frac{X_{1}+X_{2}}{X_{1}+X_{2} + X_{3}})$. Here is what I did.. $E(\frac{X_{1}+X_{2}}{X_{1}+X_{2} + X_{3}}) = E(1 ...
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Total correlation (mutual information) of exchangeable variables

The total correlation (aka multivariate mutual information) between a bunch of random variables $X_1,...,X_n \in \mathcal{X}_1 \times ... \times \mathcal{X}_n$ with joint density $p(x_1,...,x_n)$ and ...
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Estimating and testing correlation of longitudinal random variables

Each patient (indexed by $i$) contains multiple measurements of two variables $X_{i,t}$ and $Y_{i,t}$ over time $t=1, \dots, T$. For each time point $t'$, assume the correlation $\mathrm{cor}(X_{i,t'},...
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understanding exchangeability in GP regression

It is well known that exchangeability refers to the following property $p(X_1,\dots, X_n) = p(X_{\pi(1)}, \dots, X_{\pi(n)})$ for any finite $n$ and a permutation $\pi$ when we have an infinite ...
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Exchangeability and joint distribution

The definition of an exchangeabilty for a finite sequence says that, if we have random variables $X_1,\ldots,X_n$, then for each permutation $\pi: \{1,\ldots,n\}\rightarrow\{1,\ldots,n\}$, the joint ...
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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 ...
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How does one actually determine if a sequence is exchangeable

Pardon me if the question is stupid, but I don't understand how you go about proving exchangeability of a random sequence. In the case of a distribution such as the normal distribution, I get that ...
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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 ...
Md Sabbir Ahmed's user avatar
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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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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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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 $\...
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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 ...
Ruben van Bergen's user avatar
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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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Can I predict if my sequence is exchangeable? [closed]

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 distribution function $P: \mathbb{R}^2\rightarrow [0,1]$. I have the following question: Are there necessary and sufficient conditions on $P$ (or on its marginals) ensuring that $$...
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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_{(...
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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 ...
fiorenza2's user avatar
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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 ...
learning's user avatar
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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 $(X,Y,Z)$ ...
Star's user avatar
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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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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 ...
WestCoastProjects's user avatar
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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 ...
Sebastian's user avatar
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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 ...
Mr K's user avatar
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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 ...
funklute's user avatar
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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 ...
suckrates's user avatar
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3 votes
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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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