Rao-Blackwell identity for a fixed-denominator rejection estimator ($\mathbb{E}\left[\frac{M}{n}{\sum }_i{I}_i\varphi (X_i)|X\right]={\hat{\theta }}_{\mathrm{IS}}$)

For rejection indicators with conditional acceptance $f(X_i)/(Mg(X_i))$, the conditional expectation of $(M/n){\sum }_i{I}_i\varphi (X_i)$ 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.