# When to favor rejection sampling over importance sampling

I asked a question about importance vs rejection sampling: Importance sampling vs acceptance-rejection and why one would use rejection sampling when importance sampling should produce a lower variance because it doesn't outright drop any samples. A question linked was this one: What is the difference between Metropolis-Hastings, Gibbs, Importance, and Rejection sampling? which asks about a much broader range of simulation techniques and the only distinction between importance and rejection sampling I could discern (and was also pointed out) from the answer was that rejection sampling produces an i.i.d sample while importance sampling does not.

This is good to know, but I still am not able to use this information on my original question which was - what would be an actual scenario where rejection sampling would be better applicable than importance sampling. In almost all instances of simulation, I feel we end up finding some kind of average anyway. In fact, this statement was made by the professor who taught a course on simulation I took (except, he didn't say "almost").

A very specific instance when importance sampling cannot be used is the case of the pseudo-marginal Metropolis–Hastings algorithm: this is a Metropolis algorithm where the target density $$\pi(\theta)$$ is replaced with an unbiased estimator (of said density), $$\hat\pi(\theta,\xi)$$, with $$\xi\sim \varpi(\xi|\theta)$$. Ẁhile resorting to an accept-reject algorithm to produce a simulation from $$\varpi(\xi|\theta)$$ is acceptable, using instead a self-normalised importance sampling estimate of $$\pi(\theta)$$ is not valid because it breaks the unbiasedness requirement. (A simple experiment shows that the resulting Markov chain does not converge to the intended target $$\pi(\theta)$$.)
Note also that, given an accept-reject output, producing an iid sample $$x_1,\ldots,x_n$$ from an overall sample (containing rejected values) $$y_1,\ldots,y_N$$, and auxiliary uniforms, $$u_1,\ldots,u_N$$, a variance improvement over the sample average $$\frac{1}{n}\sum_{i=1}^n h(x_i)=\frac{1}{n}\sum_{j=1}^N h(y_j)\mathbb I_{u_j<\varrho(y_j)}$$ [where $$\varrho(y_j)$$ is the acceptance ratio] can be found by integrating out the $$u_j$$'s, conditional on the $$y_j$$'s and $$n$$ being accepted. The domination follows from a Rao-Blackwell argument.