## interacting particles ABC

**C**arlo Albert and Hans Kuensch recently posted an arXiv paper which provides a new perspective on ABC. It relates to ABC-MCMC and to ABC-SMC in different ways, but the major point is to propose a sequential schedule for decreasing the tolerance that ensures convergence. Although there exist other proofs of convergence in the literature, this one is quite novel in that it connects ABC with the cooling schedules of simulated annealing. (The fact that the sample size does not appear as in Fearnhead and Prangle and their non-parametric perspective can be deemed less practical, but I think this is simply another perspective on the problem!) The corresponding ABC algorithm is a mix of MCMC and SMC in that it lets a population of *N* particles evolve in a quasi-independent manner, the population being only used to update the parameters of the independent (normal) proposal and those of the cooling tolerance. Each particle in the population moves according to a Metropolis-Hastings step, but this is not an ABC-MCMC scheme in that the algorithm works with a population at all times, and this is not an ABC-SMC scheme in that there is no weighting and no resampling.

**M**aybe I can add two remarks about the conclusion: the authors do not seem aware of other works using other penalties than the 0-1 kernel, but those abound, see e.g. the discussion paper of Fearnhead and Prangle. Or Ratmann et al. The other missing connection is about adaptive tolerance construction, which is also found in the literature, see e.g. Doucet et al. or Drovandi and Pettitt.

August 27, 2012 at 5:59 am

Thanks, Ajay! Sorry I missed those refs!

August 27, 2012 at 3:13 am

Hi Christian,

I would just like to remark that Theorem 2.1 of the article is related to (in a different context), but not the same as, a result (proposition 3.1) and remark (3.1) in an article by myself, Alex Beskos and Dan Crisan:

And also some related work, Whiteley, N. (2012). Sequential Monte Carlo samplers: Error bounds and insensitivity to initial conditions. Stoch. Anal. Appl. (to appear) which discuss the behavior of SMC samplers as the number of time steps grows.

Thanks,

Ajay