  In parametric Bayesian learning, a prior is assumed on the parameter $W$
which determines the distribution of samples. In this setting, Minimum Excess
Risk (MER) is defined as the difference between the minimum expected loss
achievable when learning from data and the minimum expected loss that could be
achieved if $W$ was observed. In this paper, we build upon and extend the
recent results of (Xu & Raginsky, 2020) to analyze the MER in Bayesian learning
and derive information-theoretic bounds on it. We formulate the problem as a
(constrained) rate-distortion optimization and show how the solution can be
bounded above and below by two other rate-distortion functions that are easier
to study. The lower bound represents the minimum possible excess risk
achievable by any process using $R$ bits of information from the parameter $W$.
For the upper bound, the optimization is further constrained to use $R$ bits
from the training set, a setting which relates MER to information-theoretic
bounds on the generalization gap in frequentist learning. We derive
information-theoretic bounds on the difference between these upper and lower
bounds and show that they can provide order-wise tight rates for MER under
certain conditions. This analysis gives more insight into the
information-theoretic nature of Bayesian learning as well as providing novel
bounds.
