= Solution
Assume the draws are <independent> from $g$, that the target-weighted integrand is covered by its support, and that $\int|\theta(x)|f(x)\,dx<\infty$. Define the <importance weight> $w(x)=f(x)/g(x)$. The change-of-density identity is
$$
\mathbb E_g[w(X)\theta(X)]=\int\frac{f(x)}{g(x)}\theta(x)g(x)\,dx
=\int\theta(x)f(x)\,dx=\mu.
$$
Therefore <importance sampling> uses the unbiased estimator
$$
\boxed{\widehat\mu_g=\frac1n\sum_{i=1}^nw(x_i)\theta(x_i).}
$$
The <strong law of large numbers> gives consistency. A finite <second moment> of the weighted integrand additionally gives finite <variance> and a usual independent-sample <central limit theorem>. A useful proposal places enough <probability> where $f|\theta|$ is large and avoids tiny proposal <probability density function> there; common support alone does not guarantee finite <variance>.
Back to article page
