Rao-Blackwellizing: Is there any difference conditional on different sufficient statistics 
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*Suppose I have two different sufficient statistics $a_1$ and $a_2$ while $a_1$ summarizes information more efficient than $a_2$. 
For example, if the sample space is $\left\{y_1,y_2,y_3,y_4,y_5,y_6\right\}$, then for $a_2$, the new partition for sample space may be $\left\{r_1,r_2,r_3\right\}$ $r_1$ summarizes information from $y_1$ and $y_2$, $r_2$ summarizes information from $y_3$ and $y_4$, and so on. Then for $a_2$, the original sample space is divided into three pieces. If using $a_1$, the new sample space may be $\left\{u_1,u_2\right\}$, just two partitions. Since for $a_1$, it has fewer partitions, thus achieving data reduction more efficiently. 

*$x$ is an unbiased estimator for $\theta$. When using Rao-Blackwellizing, I get two different unbiased estimators if conditional $x$ on different sufficient statistics: $\mathbb{E}(x|a_1)$ and $\mathbb{E}(x|a_2)$.


Can I say that $V(\mathbb{E}(x|a_1)) < V(\mathbb{E}(x|a_2))$ ? If so, how to prove?
 A: If you think about where the result comes from the answer is a bit more clear.  Rao-Blackwellization is based on the fact that
\begin{align}
\text{Var}(T) &= \text{Var}[\text{E}(T \mid S)] + \text{E} [ \text{Var}(T \mid S)] .
\end{align}
So if you have two sufficient statistics $S_1$ and $S_2$ then $\text{Var}[\text{E}(T \mid S_1)] < \text{Var}[\text{E}(T \mid S_2)]$ if and only if $\text{E} [ \text{Var}(T \mid S_1)] > \text{E} [ \text{Var}(T \mid S_2)]$.  This means that, on average, conditioning on $S_1$ causes a smaller decrease in the variance of $T$, or that $S_1$ provides less information about $T$.  This is in line with the idea of $S_1$ generating a more "coarse" partitioning of the sample space than $S_2$, which is to say it achieves greater data reduction.
In general it does matter what sufficient statistic we're conditioning on.  Consider for example that the whole sample is itself a trivial sufficient statistic, but in this case conditioning doesn't actually accomplish anything.  What we want are sufficient statistics that summarize the information in the sample, ideally using a single number.
A: Rao Blackwellizing while conditioning on a sufficient statistic gives you a better estimator given that you initial estiamtor is unbiased.
However if you condition of a complete sufficient statistic then you will reach the best unbiased estimator.
There is no general rule that I know of, that specifies which sufficient statistic will give smaller variance, you will need to check the variance of the estimators
