= Bayes estimator under reciprocal weighted quadratic loss
{c}
{title2=$d_B=(\mathbb E[\Theta^{-1}\mid X])^{-1}$}
For a positive parameter $\Theta$ and loss $L(d,\Theta)=\Theta^{-1}(\Theta-d)^2$, assume $\mathbb E[\Theta\mid X]<\infty$ and $0<\mathbb E[\Theta^{-1}\mid X]<\infty$. Then
$$
\mathbb E[L(d,\Theta)\mid X]=\mathbb E[\Theta\mid X]-2d+d^2\mathbb E[\Theta^{-1}\mid X].
$$
This quadratic has the unique minimizer $d=(\mathbb E[\Theta^{-1}\mid X])^{-1}$, a harmonic rather than arithmetic <posterior mean>. Both moment conditions matter: finiteness of the inverse moment alone does not prevent the <posterior expected loss> from being infinite for every finite action.
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