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.

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