  In this paper we analyze, for a model of linear regression with gaussian
covariates, the performance of a Bayesian estimator given by the mean of a
log-concave posterior distribution with gaussian prior, in the high-dimensional
limit where the number of samples and the covariates' dimension are large and
proportional. Although the high-dimensional analysis of Bayesian estimators has
been previously studied for Bayesian-optimal linear regression where the
correct posterior is used for inference, much less is known when there is a
mismatch. Here we consider a model in which the responses are corrupted by
gaussian noise and are known to be generated as linear combinations of the
covariates, but the distributions of the ground-truth regression coefficients
and of the noise are unknown. This regression task can be rephrased as a
statistical mechanics model known as the Gardner spin glass, an analogy which
we exploit. Using a leave-one-out approach we characterize the mean-square
error for the regression coefficients. We also derive the log-normalizing
constant of the posterior. Similar models have been studied by Shcherbina and
Tirozzi and by Talagrand, but our arguments are much more straightforward. An
interesting consequence of our analysis is that in the quadratic loss case, the
performance of the Bayesian estimator is independent of a global "temperature"
hyperparameter and matches the ridge estimator: sampling and optimizing are
equally good.
