A van Trees inequality for estimators on manifolds

Van Trees’ Bayesian version of the Cramér–Rao inequality is generalised here to the context of smooth loss functions on manifolds and estimation of parameters of interest. This extends the multivariate van Trees inequality of Gill and Levit (1995) [R.D. Gill, B.Y. Levit, Applications of the van Trees inequality: a Bayesian Cramér–Rao bound, Bernoulli 1 (1995) 59–79]. In addition, the intrinsic Cramér–Rao inequality of Hendriks (1991) [H. Hendriks, A Cramér–Rao type lower bound for estimators with values in a manifold, J. Multivariate Anal. 38 (1991) 245–261] is extended to cover estimators which may be biased. The quantities used in the new inequalities are described in differential-geometric terms. Some examples are given.