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 have hit on the same question discussed in the well-known Why Most Published Research Findings Are False paper. If you do a lot of experiments as a scientific community and quite a few of the tested null hypotheses are true (i.e. people try to show a whole lot of effects that aren't really there, while some are), then "underpowered" studies are more likely to produce false positive findings than "well-powered" studies. Similarly, once one conditions on statistical significance, point estimates are biased away from where you put your null hypothesis. This bias is larger, the more underpowered a study is.
You might critique this by saying that null hypotheses are rarely exactly true, but the exact same things happen when you instead look at a set-up where many effects are very small and only a few are big.
People worry about this a lot in drug development, where large companies will run early stage proof of concept studies (you can look at those as a kind of screening tool for deciding which projects to pursue further) for many potentially promising new drugs (of which most will not have a meaningful effect on the disease of interest). It is important for these studies to not be completely underpowered, because otherwise "positive" proof of concept results will become useless as a tool for prioritizing which drugs to study further.
