Importance sampling plays an odd role in statistical computing. It’s an old-fashioned idea and can behave just horribly if applied straight-up—but it keeps arising in different statistics problems.
Aki came up with Pareto-smoothed importance sampling (PSIS) for leave-one-out cross-validation.
We recently revised the PSIS article and Dan Simpson wrote a useful blog post about it the other day. I’m linking to Dan’s post again here because he gave it an obscure title so you might have missed it.
We’ve had a bunch of other ideas during the past few years involving importance sampling, including adaptive proposal distributions, wedge sampling, expectation propagation, and gradient-based marginal optimization, so I hope we can figure out some more things.