  Van Trees inequality, also known as the Bayesian Cram\'er-Rao lower bound, is
a powerful tool for establishing lower bounds for minimax estimation through
Fisher information. It easily adapts to different statistical models and often
yields tight bounds. Recently, its application has been extended to distributed
estimation with privacy and communication constraints where it yields
order-wise optimal minimax lower bounds for various parametric tasks under
squared $L_2$ loss.
  However, a widely perceived drawback of the van Trees inequality is that it
is limited to squared $L_2$ loss. The goal of this paper is to dispel that
perception by introducing a strengthened version of the van Trees inequality
that applies to general $L_q$ loss functions by building on the Efroimovich's
inequality -- a lesser-known entropic inequality dating back to the 1970s. We
then apply the generalized van Trees inequality to lower bound $L_q$ loss in
distributed minimax estimation under communication and local differential
privacy constraints. This leads to lower bounds for $L_q$ loss that apply to
sequentially interactive and blackboard communication protocols. Additionally,
we show how the generalized van Trees inequality can be used to obtain
\emph{local} and \emph{non-asymptotic} minimax results that capture the
hardness of estimating each instance at finite sample sizes.
