= Solution
With $r$, $Z$, and $W$ as above, <independence> of the draws gives the <variance of a sample mean>:
$$
\boxed{\operatorname{Var}(\widehat p_1)=\frac1T\left[\int\frac{r(\theta)^2}{q(\theta)}\,d\theta-Z^2\right]}.
$$
This formula allows infinite <variance>. To identify the optimum, write $\pi(\theta)=p(\theta\mid\mathcal D)=r(\theta)/Z$. Then
$$
\operatorname{Var}(\widehat p_1)=\frac{Z^2}{T}\left[\int\frac{\pi^2}{q}-1\right]=\frac{Z^2}{T}\int_{\{q>0\}}\frac{(\pi-q)^2}{q}\,d\theta.
$$
The last equality uses $\int\pi=\int q=1$. Its nonnegative integrand proves
$$
\boxed{q_{\rm opt}(\theta)=\pi(\theta)=p(\theta\mid\mathcal D),\qquad \min\operatorname{Var}(\widehat p_1)=0}.
$$
Indeed, all <importance weights> then equal $Z$. This is a formal optimum: evaluating the normalized <posterior density> generally already requires the unknown <marginal likelihood>.
\b[Poor coverage of regions with substantial posterior mass produces large weights.] In particular, excessively light proposal tails, a proposal that is too narrow, or a missed mode can make $\int r^2/q$ enormous or infinite. A proposal that is unnecessarily diffuse can also waste nearly all draws on negligible <posterior probability>. Omitting target support altogether violates unbiasedness, rather than merely increasing <variance>.
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