This paper presents a lower bound, derived from the information inequality for the Bayes risk with respect to truncated priors under quadratic loss. It is discussed in cases where the regularity condition of Brown and Gajek is not always satisfied. A related result for the minimax risk is also given.
"An information inequality for the Bayes risk." Ann. Statist. 24 (5) 2288 - 2295, October 1996. https://doi.org/10.1214/aos/1069362323