Questions tagged [exchangeability]
A set of random variables is exchangeable when their joint distribution is invariant under any permutation of the random variables.
68
questions
1
vote
1answer
56 views
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 ...
3
votes
0answers
28 views
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 ...
1
vote
0answers
57 views
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 ...
0
votes
1answer
108 views
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'},...
3
votes
0answers
69 views
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 ...
6
votes
2answers
300 views
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 ...
7
votes
3answers
600 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 ...
0
votes
1answer
60 views
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 ...
1
vote
1answer
139 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 ...
6
votes
5answers
306 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
3answers
376 views
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 ...
4
votes
0answers
284 views
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 ...
1
vote
1answer
307 views
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 $\...
3
votes
2answers
217 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
0answers
90 views
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 ...
4
votes
0answers
200 views
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 ...
1
vote
0answers
39 views
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]]
...
3
votes
1answer
153 views
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
$$...
6
votes
2answers
224 views
Correlation between an observation and its rank in a random sample
Suppose $X_1,X_2,\ldots,X_n$ are i.i.d random variables with an absolutely continuous distribution.
We say the observation $X_i$ has rank $R_i$ if $$X_i=X_{(R_i)}\quad,\,i=1,2,\ldots,n,$$
where $X_{(...
2
votes
0answers
48 views
How is exchangeability related to covariate shift?
I understand that exchangeability refers to the notion that the order of data in a sequence does not affect the joint distribution of that data. In a sense, the current data we possess is from the ...
15
votes
4answers
467 views
If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$
Question
If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$.
Attempt: Please check if the below is correct.
Let ...
3
votes
0answers
96 views
From bivariate to trivariate probability distribution
Let $\mathcal{G}$ be the space of all possible bivariate probability distributions.
Let's pick a bivariate probability distribution $g\in \mathcal{G}$. Can we always find a random vector $(X,Y,Z)$ ...
8
votes
2answers
191 views
Are products of exchangeable RVs exchangeable?
Assume that $$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 ...
1
vote
1answer
497 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
1answer
114 views
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 ...
4
votes
1answer
274 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 ...
2
votes
1answer
773 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 ...
1
vote
1answer
185 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 ...
3
votes
1answer
140 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 ...
10
votes
3answers
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 ...
2
votes
1answer
288 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 ...
3
votes
1answer
76 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 ...
3
votes
1answer
46 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 ...
7
votes
2answers
421 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
votes
1answer
56 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-...
1
vote
1answer
189 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)})_{...
2
votes
0answers
259 views
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 ...
0
votes
0answers
144 views
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 ...
2
votes
2answers
55 views
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 ...
7
votes
1answer
472 views
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 ...
5
votes
1answer
444 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 ...
1
vote
1answer
186 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 ...
5
votes
2answers
836 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....
4
votes
2answers
295 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(...
2
votes
1answer
173 views
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 ...
5
votes
1answer
548 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 ...
0
votes
0answers
181 views
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)\,.
...
1
vote
2answers
2k 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 ...
4
votes
2answers
738 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
376 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 $(X_1, …, X_n)$ ...
