  The Bayesian posterior minimizes the "inferential risk" which itself bounds
the "predictive risk". This bound is tight when the likelihood and prior are
well-specified. However since misspecification induces a gap, the Bayesian
posterior predictive distribution may have poor generalization performance.
This work develops a multi-sample loss (PAC$^m$) which can close the gap by
spanning a trade-off between the two risks. The loss is computationally
favorable and offers PAC generalization guarantees. Empirical study
demonstrates improvement to the predictive distribution.
