Conditionally unbiased estimator ($E_{\theta }(\hat{\theta }\mid \mathcal{C})=\theta$)

An estimator is conditionally unbiased for a parameter if its expectation, conditional on the specified observation or selection event, equals that parameter. Unconditional unbiasedness does not imply this property after selection. Independent new-stage data and the Rao-Blackwell theorem can provide useful constructions.

A UMVCUE has minimum conditional variance among all estimators unbiased in the selected experiment for every parameter. For independent Gaussian stage estimates $A,V$ with information $I_1,J$, continuation $A\ge c$, total information $I_2=I_1+J$ and pooled estimate $T$, Rao-Blackwellization of the fresh estimate gives $T-(I_{1}s/J)\varphi ((T-c)/s)/\Phi ((T-c)/s)$, where $s^2=1/I_1-1/I_2$. The conditional family has complete sufficient statistic $T$, so the Lehmann–Scheffé theorem establishes the optimum. A conditional-likelihood bias correction does not automatically have this unbiasedness property.