  We study the distribution of the maximum likelihood estimate (MLE) in
high-dimensional logistic models, extending the recent results from Sur (2019)
to the case where the Gaussian covariates may have an arbitrary covariance
structure. We prove that in the limit of large problems holding the ratio
between the number $p$ of covariates and the sample size $n$ constant, every
finite list of MLE coordinates follows a multivariate normal distribution.
Concretely, the $j$th coordinate $\hat {\beta}_j$ of the MLE is asymptotically
normally distributed with mean $\alpha_\star \beta_j$ and standard deviation
$\sigma_\star/\tau_j$; here, $\beta_j$ is the value of the true regression
coefficient, and $\tau_j$ the standard deviation of the $j$th predictor
conditional on all the others. The numerical parameters $\alpha_\star > 1$ and
$\sigma_\star$ only depend upon the problem dimensionality $p/n$ and the
overall signal strength, and can be accurately estimated. Our results imply
that the MLE's magnitude is biased upwards and that the MLE's standard
deviation is greater than that predicted by classical theory. We present a
series of experiments on simulated and real data showing excellent agreement
with the theory.
