Bayes estimator under reciprocal weighted quadratic loss ($d_B=(\mathbb{E}[{\Theta }^{-1}\mid X]{)}^{-1}$)

For a positive parameter $\Theta$ and loss $L(d,\Theta )={\Theta }^{-1}(\Theta -d{)}^2$, assume $\mathbb{E}[\Theta \mid X]<\infty$ and $0<\mathbb{E}[{\Theta }^{-1}\mid X]<\infty$. ThenThis 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.