  We work out a version of the van Trees inequality in a Hajek--Le Cam spirit,
i.e., under minimal assumptions that, in particular, involve no direct
pointwise regularity assumptions on densities but rather almost-everywhere
differentiability in quadratic mean of the model. Surprisingly, it suffices
that the latter differentiability holds along canonical directions -- not along
all directions. Also, we identify a (slightly stronger) version of the van
Trees inequality as a very instance of a Cramer--Rao bound, i.e., the van Trees
inequality is not just a Bayesian analog of the Cramer--Rao bound. We provide,
as an illustration, an elementary proof of the local asymptotic minimax theorem
for quadratic loss functions, again assuming differentiability in quadratic
mean only along canonical directions.
