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I'm reading about the Approximate Bayesian Computation (ABC) method, and I came across two rather popular approaches:

  • Sequential Monte Carlo (SMC)

    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.

  • Population Monte Carlo (PMC)

    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?

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What I mean by this remark is that population Monte Carlo is sequential, in the sense that one constructs a weighted sample by importance sampling and utilises this sample to build a more efficient importance function for the following iteration. In the original ABC-PMC paper, we did not stress enough the fact that the target may change at each iteration, using for instance a decreasing sequence $\epsilon_t$ of tolerances.

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