Mathematics
Sufficient Statistics and the Rao-Blackwell Theorem
Quick fact
The Rao-Blackwell theorem guarantees that you can always improve an unbiased estimator—without adding bias—by averaging it over all datasets that produce the same sufficient statistic. This sometimes transforms a crude guess into the best possible unbiased estimator, with the minimal variance.
Why this is interesting
Imagine you have a raw dataset with thousands of numbers, but you only need a simple summary to estimate a population parameter. Could that summary retain all the statistical information? And could it even help you build a better estimator?