Before revision:  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  . This analysis gives more insight into the information-theoretic nature of Bayesian learning as well as providing novel bounds.
After revision:  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.