= Rao-Blackwell identity for a fixed-denominator rejection estimator
{c}
{title2=$\mathbb E\left[\frac Mn\sum_iI_i\phi(X_i)\Bigm|X\right]=\widehat\theta_{\rm IS}$}
For rejection indicators with conditional acceptance $f(X_i)/(Mg(X_i))$, the conditional expectation of $(M/n)\sum_iI_i\phi(X_i)$ given all proposals is the unnormalized importance estimator. The <Rao-Blackwell theorem> reduces variance relative to that fixed-denominator estimator. It gives no such automatic comparison with the accepted-sample mean, whose denominator is the random count.
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