Suppose a quantum state depends on unknown parameters which we want to estimate. One approach to designing good experiments is to look for the experiment which maximizes a suitable function of the Fisher information matrix for the parameters. I will combine a quantum Cramér-Rao bound due to Holevo (1980) with a classical Bayesian Cramér-Rao bound (the van Trees inequality) in order to derive asymptotic bounds, which are expected to be sharp, and which can be used to guide the choice of the experimenter. The key ingredient will be convexity properties both of the class of available information matrices, and of their inverses.