 Van Trees inequality

ID: van-trees-inequality

The van Trees inequality is a Bayesian Cramer--Rao bound. For scalar likelihood information

I (θ)

and a differentiable prior density

π

vanishing at its boundary,

\int E_{θ} (δ - θ)^{2} π (θ) d θ \geq \frac{1}{\int I ( θ ) π ( θ ) d θ + \int ( π ^{'} ( θ ) ) ^{2} / π ( θ ) d θ} .

It follows by integration by parts and the Cauchy--Schwarz inequality applied to the joint likelihood-prior score.

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