In both importance sampling as well as acceptance-rejection, we sample from some alternate distribution to simulate some expression from an original distribution of which we know the PDF. The distinction is in what we do with the samples from the alternate distribution. We can either reject them with some probability, leading to rejection sampling. Or, we can weigh them differently, leading to importance sampling.
Shouldn't it always be possible to do importance sampling when it is to do rejection sampling? If so, it seems intuitive that importance sampling should produce a lower variance than rejection since we have more freedom over the weights (as opposed to rejection where the weights are always binary). Why then would one use rejection sampling over importance sampling?
