transdimensional ABC
Another paper that got arXived recently (it’s endless, isn’t it?!) is Transdimensional ABC for inference on invasive species models with latent variables of unknown dimension, by Chkrebtii, Cameron, Campbell, and Bayne. It attracted my attention for at least two reasons: (a) it brings a new perspective on Bayesian inference in varying dimension models (or in multiple models and model comparison); (b) the application is about invasive species, as in our ABC paper on tracing pathways for the Asian beetle invasion in Europe.
After reading the paper, I however remain unconvinced that a direct duplication of Peter Green’s reversible jump MCMC algorithm is relevant in this ABC setting: this is indeed the central idea of the authors, namely to apply the reversible jump construct in an ABC-MCMC algorithm, with exactly the same validation as the usual ABC-MCMC algorithm where the indicator of a small enough distance between the observed and the simulated data acts as a (biased) estimator of the likelihood function. There is thus no doubt about the validity of the method. What leaves me somehow lukewarm about this idea is the same feature I criticised yesterday in the accelerated ABC paper by Picchini and Forman (which would also apply here), that is, the fact that the acceptance step actually occurs at the prior level, the Metropolis-Hastings acceptance probability being the ratio of the priors over the ratio of the proposals. (Plus a second acceptance step induced by the distance between the observed and the simulated data.)
The application to the invasion of European earthworms in northern Alberta is quite interesting, from the fact that those worms did not crawl their way up there but instead hitch-hiked!, to the modelling of the number of introductions by a Poisson spatial process, to the fact that the ABC algorithm can run with infinite precision! This last point makes me wonder whether or not a regular MCMC algorithm is unattainable for this problem. (However, the authors rely on a two-dimensional summary statistic for each pair (g,r), which helps in picking an ε equal to zero.) The details of the dependence of the observables on the number of earthworm introductions k are rather sketchy and I do not see why exactly the transdimensional perspective is needed. (Marginalia: I do not understand the introduction of D in the second equation of page 5, unless this is an approximation.)
Xi'an's Og 
October 28, 2013 at 4:15 am
I am a little bit confused about the paper in a couple of ways. In most applications of ABC, I would have thought that the density of the simulated data (here these are the latent variables) would be intractable. But because the authors are using local moves to update the latent variables, then these densities appear to be required in the MH ratio. So I am not sure how widely this approach could be applied in ABC.
Secondly I am a bit confused about how the latent variables are generated in algorithm 3. From the notation it appears that it comes from the full conditional (i.e. conditional on data as well). If this were possible, then ABC would not be required. I suspect that this must just be a forward simulation based on model parameters, not conditioned on data.