Van Trees inequality
= Van Trees inequality
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{wiki}
The van Trees inequality is a Bayesian Cramer--Rao bound. For scalar likelihood information $I(\theta)$ and a differentiable prior density $\pi$ vanishing at its boundary,
$$
\int\mathbb E_\theta(\delta-\theta)^2\pi(\theta)\,d\theta
\ge
\frac1{\int I(\theta)\pi(\theta)\,d\theta
+\int(\pi'(\theta))^2/\pi(\theta)\,d\theta}.
$$
It follows by integration by parts and the Cauchy--Schwarz inequality applied to the joint likelihood-prior score.