This paper presents lower bounds, derived from the information inequality, for the Bayes risk under scaled quadratic loss. Some numerical results are also presented which give some idea concerning the precision of these bounds. An appendix contains a proof of the information inequality without conditions on the estimator. This result is a direct extension of an earlier result of Fabian and Hannan.
"Information Inequalities for the Bayes Risk." Ann. Statist. 18 (4) 1578 - 1594, December, 1990. https://doi.org/10.1214/aos/1176347867