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Consider a multivariate logistic-normal variable $z \sim \mathcal {LN}(\mu,{\Sigma})$, where ${\Sigma}$ is and $n$-by-$n$ positive definite matrix. I mean, for $x = (x_1,\ldots,x_n)\sim \mathcal N(\mu,\Sigma)$ define the vector $z = (z_1,...,z_n, z_{n+1})$ by $$z_j = \dfrac{\exp(x_j)}{\sum_{l=1}^{n+1} \exp(x_l)}, $$ with $x_{n+1} := 0$.

Observations: It is well-known that in general, the moments of $z$ admit not a closed-form solution. There are some trivial exceptions though. For example if $n = 1$ and $\mu = 0$, then it's easy to see that probability density of $z$ is symmetric about the point line $z = 1/2$ and $z$ has mean $1/2$, irrespective of the variance $\sigma^2$ of $x$.

Question 1: In a general dimension $n \ge 2$, if $\mu = 0 \in \mathbb R^n$, what can be said about the expected value $\mathbb E[z]$ of $z$ ?

A completely blind guess is the centroid $(1/n,\ldots,1/n)$ of the $n$-dimensional simplex (at least in the spherical case $\Sigma = \operatorname{I}$).

Question 2: Say I knew before hand that $\mathbb E [z] = (1/n,\ldots,1/n)$. What can I say about the mean $\mu$ of the underlying Gaussian ?

Question 3: If $\mu = \epsilon^{-1}\tilde{\mu}$, and $\Sigma = \epsilon^{-2}\tilde{\Sigma}$, what is (an approximation of) the value of the limit $\lim_{\epsilon \rightarrow 0^+}\mathbb E[z]$ (as a function of $\tilde{\mu}$ and $\tilde{\Sigma}$)?

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Take $n=2$ as an example. If $\mu={\bf 0}$, the density of $Z$ is $$ \frac{1}{|2\pi\Sigma|^{1/2}z_1z_2z_3} exp\{ -\frac{1}{2} {\bf z^T_{-3}} \Sigma^{-1}{\bf z_{-3}}\}, $$

where ${\bf z_{-3}} = (log(z_1/z_3), log(z_2/z_3))^T$ and $\sum_{i=1}^3 z_i =1$.

For question #1, if $\Sigma = I_2$, $E({\bf z})$ is not $(1/3,1/3,1/3)^T$ by numerical evaluation, and $E(z_1)=E(z_2)\ne E(z_3)$. Note the exponent part of the density $exp\{-1/2 [(log(z_1/z_3))^2+ (log(z_2/z_3))^2]\}$ is not symmetric for $z_1,z_2,z_3$. However, it is symmetric for $z_1,z_2$. To the best of my knowledge, there is no closed form for the expectation.

For question #2, to make the expectations equal for each component, one sufficient condition is to make the exponent part in the density symmetric for $z_1,z_2,z_3$. Note that if $$ \Sigma = c\pmatrix {1&0.5\\0.5&1}, c>0, $$ the exponent part becomes proportional to $$ (log(z_1/z_3))^2+ (log(z_2/z_3))^2-log(z_1/z_3)log(z_2/z_3) = [log(z_1)]^2+[log(z_2)]^2+[log(z_3)]^2-log(z_1)log(z_2)-log(z_1)log(z_3)-log(z_2)log(z_3), $$ which is symmetric for $z_1,z_2,z_3$. Thus the expectation $E(z_1)=E(z_2)= E(z_3)=1/3$. This is generalizable to any $n \ge 2$. If $\mu=0$ , the diagonal elements of $\Sigma$ are the same and all the off-diagonal elements of $\Sigma$ are half of the diagonal, the density would be symmetric for $z_1,...,z_{n+1}$, which implies $ E(\bf{z}) = \bf{1/(n+1)}$. I am not sure if this is a necessary condition and I am interested if there exist other examples either when $\mu=0$ or $\mu \ne 0$.

I do not understand the idea of question #3. Are you thinikng what the expectation would be when $\mu \rightarrow \infty$ ?

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  • $\begingroup$ Thanks for the response. Those are some really nice ideas. About Q#3, not quite. For example, take $c = \epsilon^{-1}$ and $\mu = \epsilon^{-1}(a,b)$, for some real scalars $a$ and $b$ (e.g take $a=b=0$ for purpose of illustration) in your example above and let $\epsilon \rightarrow 0^+$. I'm interested in the limiting value of expectations $\mathbb E[z_1]$ and $\mathbb E[z_2]$ $\endgroup$ – dohmatob Dec 8 '17 at 15:11
  • $\begingroup$ I think it depends on $a,b$ and the variance. If variance is equal and $c=\epsilon^{-2}$, then $E(z_i)$ with highest mean $\mu_i$ will tend to 1 no matter what $\sigma_{ij}$ is. If variances are not equal, I am not sure. I guess it will depend on $\sigma_{ij}, \mu$. $\endgroup$ – Statisfun Dec 8 '17 at 16:23
  • $\begingroup$ Ya sure, i seek the limit as a function of $\tilde{\mu}$ and $\tilde{\Sigma}$ as in the statement of the question. I'll make this more explicit in the post. $\endgroup$ – dohmatob Dec 8 '17 at 16:38

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