With respect to your second point that “the rejection takes place without incorporating the information contained in the data”, this is a fundamental part of why the 1-hit algorithm analyzed with Krzysztof Łatuszyński has conditions on the “interplay between the likelihood and the prior-proposal pair”. This was mentioned in ABC in Rome (talk is available below) and we are more explicit in Remark 4 of http://arxiv.org/abs/1210.6703 where we point out that the Metropolis-Hastings counterpart would not require such conditions as it always evaluates the product of the prior and the likelihood. In particular, for a random walk proposal, ABC-MCMC can be in trouble when the prior and the likelihood are in extreme disagreement but their product has exponential or lighter tails.

https://sites.google.com/site/approxbayescompinrome/program/speakers

Of course, while this is an identifiable issue, it is not clear (at least to me) if people will often have priors and likelihoods that conflict enough for this to be important in practical applications.

I remember briefly discussing with Umberto in Rome about “early rejection”, as during my talk I had (perhaps wrongly) just assumed that people implementing Marjoram et al.’s ABC-MCMC kernel would use early rejection, at least in models where simulating the data is expensive.

]]>Actually no (I saw Stephen Walker talk about that one in Duke, where it was not as well received as I would’ve expected [although there are big cultural differences between countries…]). This one was much more in the spirit of estimating discrete HMMs, but with the focus on estimating loss functions rather than parameters or smoothing (it was a fowards-backwards algorithm, although I can’t seem to find a paper).

Incidentally, Chris H gave a great talk here last weekend at our annual workshop on new work that really extended the paper you linked. A pretty facile summary would be “All models are wrong, so what are we going to do about it?”!!!

]]>At the risk of making a “dangerous” suggestion, if one knew how imprecise the ABC posterior was in the first place, then simulating “enough” extra data could balance the precision of the ABC posterior to be close to that of the true posterior.

No I have no idea how to do this. A project for someone :-)

]]>Thanks, Chris. This subsampling idea sounded a wee bit dangerous to me…

]]>Dan, do you mean this paper? This may prove too much of an approximation for mainstream statisticians. (Or not, if proven to be the only manageable solution!) Btw, it was great seeing you last week in Warwick!

]]>Thanks, Scott: given that the ratio of the kernels appears in the acceptance probability, you would need an upper bound on the ratio of the kernels, don’t you? Which would be either unrealistic or highly conservative….

]]>you never know, maybe I booked an helicopter…?!

]]>There are 2 accelerations involved in the first linked paper (with J. Forman): the first one is due to an “early rejection” approach and the second one due to a “subsampling” procedure. I want to emphasize (although Christian has been already explicit in this sense) that early-rejection is due to a previous paper (denoted as “earlier ABC work” in the post), and early-rejection is completely independent of “subsampling”, i.e. it can be applied regardless of subsampling. The subsampling acceleration is not considered/suggested as a general procedure (of course) but it makes sense in the specific context of the proposed large-data application, where observations follow a fairly regular pattern.

As pointed out by Scott (thanks!), I do agree that subsampling must introduce a loss of precision as “subsampled statistic is able to reproduce the observed statistic (itself subsampled or otherwise) from a wider range of parameter values.” but also that, following Christian, in the end there are so many levels of approximation in ABC that from a practical point of view this might be a minor issue for the considered application (although relevant for the theory).

I would also like to comment on Christian’s remark that early-rejection ” always speeds up the algorithm”: as pointed-out in the earlier paper (the one without subsampling) this is not necessarily the case, under specific circumstances, for example when a uniform kernel is taken and when uniform priors are placed on ALL unknowns. In such case the acceptance-ratio for the early-rejection step would equal 1 and therefore the simulated uniform r.v. cannot exceed such ratio (thus early-rejection never takes place). In my experiments (and in my abc-sde package — btw the correct link is http://sourceforge.net/projects/abc-sde/) the ABC-MCMC targets the augmented posterior for (\theta,\varepsilon) and an exponential prior is considered for the ABC tolerance \varepsilon, so this extreme situation does not occur even when a uniform prior is set on \theta.

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