= Solution
An <independence sampler> proposes from a fixed density $g(\theta')$, independent of the current state. Its acceptance probability is
$$
\boxed{\alpha(\theta,\theta')=\min\left\{1,
\frac{\pi(\theta')g(\theta)}{\pi(\theta)g(\theta')}\right\}.}
$$
If $g$ is close to the target, proposals can make large efficient moves and cross separated modes; $g=\pi$ gives acceptance one and independent samples. On the other hand, a good global approximation is required. A proposal with tails too light can leave the chain trapped at states having very large importance weight $\pi/g$, since most outgoing proposals then have tiny acceptance probabilities.
For perspective, if the normalized target satisfies $\pi\le Mg$, the accepted transition density is at least $\pi(\theta')/M$, because both entries in $\min\{g(\theta'),\pi(\theta')/w(\theta)\}$ have that lower bound, where $w=\pi/g\le M$. This yields a uniform refresh component. Thus a well-designed independence proposal can mix rapidly, whereas a poor global proposal can be much worse than a locally tuned random walk. The tradeoff concerns effective samples per computation, not acceptance rate alone.
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