Power paradox: overestimated effect size in low-powered study, but the estimator is unbiased If we have an underpowered study but manage to reject the null hypothesis, anyway, it makes sense to wonder if we have overestimated the effect size.
However, such a concern seems unwarranted if we use an unbiased estimator of the effect size (such as $\bar X_1-\bar X_2$ to estimate $\mu_1-\mu_2$).
At the same time, there is an appeal to speculating that we must have lucked into a particularly large observed effect to have managed to reject despite the low power.
This seems like a paradox. Does it have a resolution?

You may have an estimator $\hat\theta$ that is (unconditionally) unbiased for its target: $\mathbb{E}(\hat\theta)=\theta$. The absolute value of the estimator $|\hat\theta|$ may also be (unconditionally) unbiased for the absolute value of the target: $\mathbb{E}(|\hat\theta|)=|\theta|$. (The absolute value rather than the raw value is relevant when considering effect size.)
However, once you condition on statistical significance of the estimate, the absolute value of the conditional estimator will generally no longer be unbiased for the absolute value of the target: $\mathbb{E}(|\hat\theta|\mid \hat\theta\text{ is stat. signif. at }\alpha\text{ level})\neq|\theta|$.
(I had struggled with a similar question over here: Understanding Gelman & Carlin "Beyond Power Calculations: ..." (2014). The issue was not really the essence but rather presentation. In the beginning it was not immediately clear to me that Gelman & Carlin were actually conditioning on statistical significance.)
