Rao Blackwellization
Average an unbiased estimator within sufficient-count groups and compare its exact variance before and after conditioning.
Free theory articleAverage an unbiased estimator within sufficient-count groups and compare its exact variance before and after conditioning.
Concept & interpretation
The Rao-Blackwell theorem provides a constructive method for improving an unbiased estimator with finite variance. If you have a naive unbiased estimator T, and a sufficient statistic S for the parameter, conditioning the naive estimator on the sufficient statistic creates a new estimator E[T|S] which is also unbiased and has a variance less than or equal to the original estimator.
Experiment
You can interactively explore these estimators across the full sampling space using the Rao Blackwellization module.
- Set the Bernoulli probability to 0.25. The target r = p(1-p) is 0.1875.
- Set the Observed pair to 10. The naive estimator T evaluates to 1, while the sufficient count S evaluates to 1.
- Notice the improved conditional estimator E[T|S] evaluates to exactly 0.5.
- Observe the variance comparison across the full 4-outcome law. Both sampling expectations equal the target 0.1875. The naive variance is 0.15234375, while the improved variance drops to 0.05859375.
- Change the Observed pair to 00. For this observation, S=0, so the conditional estimator evaluates to 0. However, the theoretical variances remain the same, because they evaluate the estimator performance across all possible samples, not just the currently observed pair.