Intuitive and Formulaic Justification for the Rao-Blackwell Theorem I've tried looking online and I can't seem to grasp the Rao-Blackwell theorem.
Could someone please give an intuitive explanation backed up by formulae.
 A: First, let's recall what a sufficient statistic is: intuitively it captures all the information about a parameter there is in the sample. In other words, if we use the original sample, rather than the statistic we would not be able to estimate our parameter any better.
To formalize these two claims we can use factorization criterion to back up the first, and a Cramer-Rao lower bound for the second. 
Rao-Blackwell theorem tells you that you can always improve (or an least not do worse) on your estimator's quality (measures by variance) if you condition your estimator on the sufficient statistic. Because this statistic provides all the information about this parameter from the sample, it contains all you know about the parameter. By conditioning you use all this available information. That's why it is intuitive, it should be the best (where by best we mean smallest variance and since it is unbiased also smallest mean squared error). 
This is a chatty explanation but you asked for intuition not hard proofs:)
Hope this helps!
A: An estimator $\hat{\theta}$ of $\theta$ may (correctly) depend on features of the data that provide information on $\theta$ (i.e. depend on the sufficient statistic $T(X)$), but it may also, in general, depend on features that do not provide information on $\theta$. The latter dependence is a source of variance that we'd like to eliminate -- there is no reason why two samples, with the same value of the sufficient statistic, should provide different estimates for $\theta$.
So a simple way to eliminate this variance is to just average the estimator over all samples with the same value of $T(X)$. This gives rise to a new estimator $E(\hat{\theta}|T(X))$, known as the Rao-Blackwellization.
The proof of the theorem is very analogous to the intuition, in that we simply do an ANOVA of the variance in $\hat{\theta}$:
$$\mathrm{Var}(\hat{\theta})=\mathrm{E}(\mathrm{var}(\hat{\theta}|T))+\mathrm{var}(\mathrm{E}(\hat{\theta}|T))$$
The latter is the variance in the Rao-Blackwellized estimator while the former is the variance we want to eliminate.
See Sufficient statistics and the Rao-Blackwell theorem for some tangential Bayesian explanation of sufficient statistics, if it is not clear why we want to eliminate this source of variance.
