  We investigate the cold posterior effect through the lens of PAC-Bayes
generalization bounds. We argue that in the non-asymptotic setting, when the
number of training samples is (relatively) small, discussions of the cold
posterior effect should take into account that approximate Bayesian inference
does not readily provide guarantees of performance on out-of-sample data.
Instead, out-of-sample error is better described through a generalization
bound. In this context, we explore the connections between the ELBO objective
from variational inference and the PAC-Bayes objectives. We note that, while
the ELBO and PAC-Bayes objectives are similar, the latter objectives naturally
contain a temperature parameter $\lambda$ which is not restricted to be
$\lambda=1$. For both regression and classification tasks, in the case of
isotropic Laplace approximations to the posterior, we show how this
PAC-Bayesian interpretation of the temperature parameter captures the cold
posterior effect.
