Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 40 / 4 / ii / Solution

Write $f=f_0/C_f$ and $g=g_0/C_g$, where $f_0,g_0$ are known nonnegative kernels. If both constants are known, the ordinary weight is $(C_g/C_f)f_0/g_0$. If their ratio is unknown, use self-normalized importance sampling:Indeed, under the normalized proposal law,The strong law of large numbers makes their empirical ratio converge to $\mu$, provided the numerator is absolutely integrable and $0<C_f,C_g<\infty$. Multiplicative constants cancel, whether the unknown one belongs to the target, proposal, or both. Unlike the estimator in part (i), this ratio is generally biased at finite sample size. Exact sampling from $g$ is still presumed; knowing only an unnormalized proposal does not itself supply a sampler.