The Blackwell-Rao Theorem

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The Rao‑Blackwell theorem claims that replacing an estimator by its conditional expectation given a sufficient statistic strictly lowers mean‑square error (and any convex loss); proof uses Fisher‑Neyman factorization, Jensen and the law of total expectation. Example: for Bernoulli T = sum xi the MLE S/n is biased, S/(n−1) unbiased, and S/(n+1) attains smaller MSE. Application: perturb sufficient statistics (e.g. the X'X matrix) for differential privacy — noise matters for small n but vanishes as sample size grows.

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1. 0:00Introduction
2. 0:45BlackwellRao Theorem
3. 3:55BlackwellRao PhD
4. 9:38Motivation
5. 22:39Black Hole Theorem
6. 27:26Application