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Let a $d-\text{dimensional}$ Wishart random variable with $\nu$ degrees of freedom $\Sigma$ be distributed according to $\mathcal{W}(\Sigma|\Sigma_0, \nu) \propto |\Sigma|^\frac{\nu-d-1}2\exp{(-\frac12\text{tr}\ \Sigma_0^{-1}\Sigma)}$.

What is the expected value of $\text{chol} \ \Sigma$?

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    $\begingroup$ Not an answer, but a Wishart is kind of a matrix valued Gamma, and chol is kind of a matrix square root, so the answer might be related to a generalized gamma, en.wikipedia.org/wiki/Generalized_gamma_distribution. See en.wikipedia.org/wiki/Gamma_distribution#Related_distributions: "If $X \sim Gamma(k, \theta)$, then $\sqrt{X}$ follows a generalized gamma distribution ..." But its moments do not really suggest a simple answer in the matrix-valued case. $\endgroup$ Sep 24, 2021 at 14:30
  • $\begingroup$ Interesting. Combined with Bartlett Decomposition (en.wikipedia.org/wiki/…), I may be able to reach an answer, since the random part of the decomposition only depends on normals and square roots of gammas (chi2), the latter part of which you just pointed toward a solution (!) and the generalized gamma does have an analytical expression for expected value! If you write up the formal answer, I'll give you the bounty. $\endgroup$
    – user27886
    Sep 24, 2021 at 18:02

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In 1933 Bartlett described the Wishart distribution in terms of the distribution of the factors after Cholesky decomposition. I can not read the original source (On the theory of statistical regression) but this 'Bartlett decomposition' has been described elsewhere many times. See for instance the Wikipedia article on the Wishart distribution, which describes it as:

$$W \sim LAA^TL$$

with $L$ the cholesky factor of the $\Sigma_0$ matrix. And $A$ a triangular matrix that is standard normal distributed for the non-diagonal elements and the root of a chi-square distributed variable for the diagonal elements (which is just a chi distributed variable).

The elements of the cholesky factor $LA$ is then a weighted sum of normal distributed variables and a chi distributed variable. To compute the mean of this sum you take the sum of the means of the individual variables. For the normal distributed variables the mean is obvious. For the chi distributed variable you can compute it using a ratio of gamma functions.

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