PAC-Bayesian Theory Meets Bayesian Inference

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.Comment: Published at NIPS 2015 (http://papers.nips.cc/paper/6569-pac-bayesian-theory-meets-bayesian-inference

The fine print on tempered posteriors

We conduct a detailed investigation of tempered posteriors and uncover a number of crucial and previously undiscussed points. Contrary to previous results, we first show that for realistic models and datasets and the tightly controlled case of the Laplace approximation to the posterior, stochasticity does not in general improve test accuracy. The coldest temperature is often optimal. One might think that Bayesian models with some stochasticity can at least obtain improvements in terms of calibration. However, we show empirically that when gains are obtained this comes at the cost of degradation in test accuracy. We then discuss how targeting Frequentist metrics using Bayesian models provides a simple explanation of the need for a temperature parameter $\lambda$ in the optimization objective. Contrary to prior works, we finally show through a PAC-Bayesian analysis that the temperature $\lambda$ cannot be seen as simply fixing a misspecified prior or likelihood