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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questions
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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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1answer
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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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2answers
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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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0answers
104 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 ...
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Mixture of $K$ components
Consider a random vector
$$
X\equiv
\begin{pmatrix}
X_1\\
X_2\\
X_3
\end{pmatrix}
$$
with pdf $$f(x)=\overbrace{\sum_{k=1}^ K \frac{1}{K} f_k(x)}^{\text{finite mixture}}$$ and $\forall k=1,...,K$ $...
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0answers
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Can I predict if my sequence is exchangeable?
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]]
...
1
vote
1answer
59 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 probability distribution $P: \mathbb{R}^2\rightarrow [0,1]$. I have the following question:
Are there necessary and sufficient conditions on the CDF associated with $P$ (joint or ...
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votes
1answer
118 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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0answers
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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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0answers
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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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2answers
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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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1answer
124 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 ...
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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 ...
3
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1answer
126 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
361 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
102 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
111 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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2answers
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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
116 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
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1answer
61 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
40 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 ...
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2answers
243 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
49 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-...
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1answer
67 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)})_{...
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0answers
134 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 ...
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0answers
142 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
46 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 ...
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1answer
329 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 ...
4
votes
1answer
350 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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vote
1answer
151 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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2answers
546 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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2answers
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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:
$$p(...
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1answer
142 views
Estimating Degrees of Freedom from Non-IID samples
Suppose I have some continuous data that's distributed according to
$$x_i \sim p(X|\theta)$$
for some unknown parameters $\theta$. We may draw some $N$ samples from density $p$. Then we can say ...
4
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1answer
387 views
Why are words in a document for bag-of-words model exchangeable but not independent?
I've been watching a talk (section between 07:20-08:00) given by Michael Jordan and I'm getting confused between independence and exchangeability.
He says that
"If we have a document and you ...
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0answers
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How the De Finetti's Representation Theorem works in this case?
For the special case of infinite sequence of $\{0,1\}$ valued random variables the theorem is stated as
$$
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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2answers
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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 ...
4
votes
2answers
369 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
264 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 $(X_1, …, X_n)$ ...
3
votes
1answer
154 views
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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1answer
71 views
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 ...
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2answers
880 views
intuition behind exchangeability property, and its use in statistical inference
I'm reading "Bayesian Data Analysis" by Gelman et al., and I encountered this exchangeability property: $\{X_n\}_{n \in N}$ is exchangeable if $F_{X_1,\ldots,X_n}(x_1,\ldots,x_n)$ is symmetric in its ...
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Deeper proof of common expected value of random convex weights
$\newcommand{\E}{\mathbb{E}}$Let a finite collection of exchangeable random variables $X_1,...X_n$ (some authors would call this collection "interchangeable" since it is finite, reserving "...
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2answers
463 views
GEE model returns GLM results
I have simulated some longitudinal data with 100 subjects and 7 measurements per subject. My data has random intercept which will induce "exchangeable" correlation matrix. My goal is to fit two ...
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1answer
1k views
Fixed Regressor Conspiracy and Connection to Exchangeability
In simple regression model regressors are treated as fixed rather than stochastic. Whoever picks the experimental values for the regressors, decides in which frequency to include each value. This can ...
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0answers
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When is an ellipsoidally (elliptical) distributed random variable spherically symmetric?
If $Y$ is ellipsoidally distributed, and $\mu_y \propto 1_p$ and $\sum_y = \sigma^2 I$ is $Y$ spherically symmetric? EDIT: added - Is it exchangeably distributed? Please give a proof not a yes/no.
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1answer
444 views
How to test the exchangeability of data?
I have implemented a constraint-based random number generator producing 3 columns with the constraint that each row sums to 1:
...
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1answer
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What is the intuition behind exchangeable samples under the null hypothesis?
Permutation tests (also called a randomization test, re-randomization test, or an exact test) are very useful and come in handy when the assumption of normal distribution required by for instance, <...
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0answers
136 views
Exchangeable Processes over the Simplex
You are likely all familiar with Polya Urn process. I initially start with an urn containing $b$ black balls and $w$ white balls. At each step, I sample a black ball with probability $\frac{b}{b+w}$ ...
