PAC-Bayesian Theory Meets Bayesian Inference
This work provides a theoretical link between PAC-Bayesian and Bayesian frameworks, which is incremental for researchers in statistical learning theory.
The paper establishes a connection between frequentist PAC-Bayesian risk bounds and Bayesian marginal likelihood, showing that minimizing these bounds for negative log-likelihood loss maximizes the marginal likelihood, offering an alternative to Bayesian Occam's razor under i.i.d. data assumptions, and validates this on Bayesian linear regression tasks.
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.