# 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 ...
1 vote
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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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### 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 ...
145 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 ...
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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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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 ...