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What is the distribution of $(X_1, ..., X_n)^T \sim \mathcal{N}(\mu, \Sigma)$ given that $X_1 = ... = X_n$?

Jannis
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  • Would $\Sigma$ be 1 everywhere? – User0 Jan 22 '23 at 17:13
  • @user0 No, $\Sigma$ is an arbitrary positive semidefinite matrix – Jannis Jan 22 '23 at 17:14
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    Express your question in a standard matrix form and apply the formulas of https://stats.stackexchange.com/questions/30588. – whuber Jan 22 '23 at 17:43
  • @whuber While I think this is a good direcction, it's not very clear how to apply it because the standard formulas for Gaussian conditional distributions only apply if we marginalize out a subset of variables which is not the case here. – Jannis Jan 22 '23 at 17:57
  • This is one of the simplest cases where you are marginalizing over everything except one variable. How to apply it is crystal clear: you need to partition $\Sigma$ and $\mu$ appropriately and then just plug the pieces into the formula. How you compute that partition of $\Sigma$ will vary according to its structure, which likely was the motivation for the initial comment by @user0. – whuber Jan 22 '23 at 18:02
  • @whuber Could you elaborate on how to partition the matrix? All the structure it has is that it is positive semidefinite. I feel like some reparametrization would be necessary. How would that work? – Jannis Jan 22 '23 at 18:28
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    The conditional distribution will be at most one-dimensional, so you will want to change your variables anyway. An attractive set to consider is to formulate the question in terms of $Y_i = X_i - X_n,$ $i=1,2,\ldots, n-1,$ and $Y_n = X_n.$ The condition $X_1=\cdots=X_n$ would be expressed as $(Y_1,\ldots,Y_{n-1})=(0,\ldots,0).$ Because $X_i=Y_n+Y_i=Y_n$ when $Y_i=0,$ finding the conditional distribution of $Y_n$ will do the trick. – whuber Jan 22 '23 at 18:45

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