Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 40 / 2 / a / ii

Write $f=\tilde{f}/Z_f$ and $g=\tilde{g}/Z_g$. If the constants needed to evaluate $f/g$ are unavailable, use the self-normalized importance sampling estimatorThe unknown common factor $Z_g/Z_f$ cancels. Under full target probability support and integrability, the numerator and denominator sample means converge to $\int \tilde{f}\theta /Z_g$ and $\int \tilde{f}/Z_g$, respectively, whose ratio is $\mu$. The same construction applies if only one normalization is unknown by taking the other probability density function already normalized. Sampling from $g$ itself is assumed available, even when its normalization is not explicitly known.

This ratio estimator is generally biased at finite $n$, unlike the ordinary normalized-weight estimator. Also, its variance is not the formula in the next part: that formula concerns the ordinary unbiased estimator. Full probability support for $f$ is needed here because the denominator estimates the entire target normalizing integral, not just the probability support of $f\theta$.