I'm reading about the Approximate Bayesian Computation (ABC) method, and I came across two rather popular approaches:
methodology to sample sequentially from a sequence of probability distributions that are defined on a common space, each distribution being known up to a normalizing constant.
principle consists of iterated generations of importance samples, with importance functions depending on the previously generated importance samples.
Reading these descriptions, they don't look very similar. But I also came across an entry in Xi'an's blog where these two approaches seem to be treated as the same thing:
astroABC: ABC SMC sampler for cosmological parameter estimation: (...) The version of ABC implemented there is “our” ABC PMC...
Where, if I understand correctly, Xi'an writes about an ABC-SMC sampler that is an implementation of his ABC-PMC sampler.
So the question is: are these two approaches equivalent?