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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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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Understanding Exchangability in Bayesian Hierarchical Models

I have a group of $k$ experiments (indexed by $j=1...k$) and each experiment $j$ produces a set of $n_j$ datapoints denoted as $y_{ij}$ such that $i\in\{1,...,n_j\}$. Meanwhile, $y_{ij}$ is ...
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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 ...
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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 ...
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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 ...
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15 votes
4 answers
567 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 ...
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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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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 ...
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3 votes
1 answer
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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 ...
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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 ...
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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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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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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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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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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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1 answer
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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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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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7 votes
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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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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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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)})_{...
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Expected value of maxima of dependent random variables

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 ...
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How to regularize when facing exchangeable vectors in deep learning?

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 ...
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2 votes
2 answers
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Estimating number of types of things, from many observations

I go on a safari. Each day, I see various animals, and a keep track of all of my observations. Assume that each time I make an observation I see a single animal. At one point, I've made a total of ...
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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 ...
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