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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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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1answer
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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
$$...
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1answer
133 views
Correlation between an observation and its rank in a random sample
Suppose $X_1,X_2,\ldots,X_n$ are i.i.d random variables with an absolutely continuous distribution.
We say the observation $X_i$ has rank $R_i$ if $$X_i=X_{(R_i)}\quad,\,i=1,2,\ldots,n,$$
where $X_{(...
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How is exchangeability related to covariate shift?
I understand that exchangeability refers to the notion that the order of data in a sequence does not affect the joint distribution of that data. In a sense, the current data we possess is from the ...
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If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$
Question
If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$.
Attempt: Please check if the below is correct.
Let ...
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From bivariate to trivariate probability distribution
Let $\mathcal{G}$ be the space of all possible bivariate probability distributions.
Let's pick a bivariate probability distribution $g\in \mathcal{G}$. Can we always find a random vector $(X,Y,Z)$ ...
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Are products of exchangeable RVs exchangeable?
Assume that $$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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1answer
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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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1answer
188 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 ...
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1answer
577 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 ...
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1answer
141 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 ...
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1answer
126 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 ...
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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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1answer
196 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 ...
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1answer
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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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1answer
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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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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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1answer
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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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2answers
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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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1answer
399 views
Spherical symmetry: a generalization of exchangeability?
In Kingman, J. (1972) On Random Sequences with Spherical Symmetry. Biometrika, 59(2), 492-494., the author states a theorem that, in his words, "has a close resemblance to de Finetti's theorem on ...
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173 views
Sharing of information and borrowing of strength. What models are there
I am currently working on identifying all the methods that have been implemented in all kinds of disciplines in order to borrow strength. If you have ever used, or came across such methods i would be ...
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692 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....
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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(...
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1answer
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Estimating Degrees of Freedom from Non-IID samples
Suppose I have some continuous data that's distributed according to
$$x_i \sim p(X|\theta)$$
for some unknown parameters $\theta$. We may draw some $N$ samples from density $p$. Then we can say ...
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1answer
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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 ...
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How the De Finetti's Representation Theorem works in this case?
For the special case of infinite sequence of $\{0,1\}$ valued random variables the theorem is stated as
$$
Pr(x_1, \ldots, x_n) = \int_0^1 p^{(\sum_{i=1}^n x_i)}[1-p]^{(n - \sum_{i=1}^n x_i)} dQ(p)\,.
...
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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 ...
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2answers
544 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 ...
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1answer
325 views
Notion of “the same” distribution in definitions of “iid” and “exchangable”
Schervish's (1995) Theory of Statistics defines exchangeability like this (p. 7):
A finite set $X_1, …, X_n$ of random quantities is said to be exchangeable if every permutation of $(X_1, …, X_n)$ ...
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When are observations not weakly exchangeable?
In the book "Common errors in statistics" https://www.amazon.com/Common-Errors-Statistics-Avoid-Them/dp/1118294394, I read the following statement
Permutation tests only yield exact significance ...
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Simple question on exchangeability
We say that $(x_1,x_2,\dots)$is an infinitely exchangeable sequence of random variables iff for any permutation $\pi$, $p(x_1,\dots,x_n)=p(x_{\pi(1)},\dots,x_{\pi(n)})$.
Let $(x_1,x_2,\dots)$ be an ...
