Theorem. Unbiasedness of the paired score estimator [ftip-00FF]
✍️source

    Under the iid coupling in Definition [ftip-00FE], \(\mathbb E[\widehat \Delta _m]=J_{\mathsf Q}(M)-J_{\mathsf Q'}(M')\).

    Proof. 

      Linearity of expectation gives the displayed identity.

      \[ \mathbb E[\widehat \Delta _m] =m^{-1}\sum _j\left (\mathbb E[s_M(Z_j)]- \mathbb E[s_{M'}(Z'_j)]\right ). \]

      The coupling fixes the joint sampling but does not change either marginal, so the two terms are the stated expectations.