  We exhibit a strong link between frequentist PAC-Bayesian risk bounds and the
Bayesian marginal likelihood. That is, for the negative log-likelihood loss
function, we show that the minimization of PAC-Bayesian generalization risk
bounds maximizes the Bayesian marginal likelihood. This provides an alternative
explanation to the Bayesian Occam's razor criteria, under the assumption that
the data is generated by an i.i.d distribution. Moreover, as the negative
log-likelihood is an unbounded loss function, we motivate and propose a
PAC-Bayesian theorem tailored for the sub-gamma loss family, and we show that
our approach is sound on classical Bayesian linear regression tasks.
