## ABC by population annealing

**T**he paper “Bayesian Parameter Inference and Model Selection by Population Annealing in System Biology” by Yohei Murakami got published in PLoS One last August but I only became aware of it when ResearchGate pointed it out to me [by mentioning one of our ABC papers was quoted there].

“We are recommended to try a number of annealing schedules to check the influence of the schedules on the simulated data (…) As a whole, the simulations with the posterior parameter ensemble could, not only reproduce the data used for parameter inference, but also capture and predict the data which was not used for parameter inference.”

Population annealing is a notion introduced by Y Iba, the very same IBA who introduced the notion of population Monte Carlo that we studied in subsequent papers. It reproduces the setting found in many particle filter papers of a sequence of (annealed or rather tempered) targets ranging from an easy (i.e., almost flat) target to the genuine target, and of an update of a particle set by MCMC moves and reweighing. I actually have trouble perceiving the difference with other sequential Monte Carlo schemes as those exposed in Del Moral, Doucet and Jasra (2006, Series B). And the same is true of the ABC extension covered in this paper. (Where the annealed intermediate targets correspond to larger tolerances.) This sounds like a traditional ABC-SMC algorithm. Without the adaptive scheme on the tolerance ε found e.g. in Del Moral et al., since the sequence is set in advance. [However, the discussion about the implementation includes the above quote that suggests a vague form of cross-validated tolerance construction]. The approximation of the marginal likelihood also sounds standard, the marginal being approximated by the proportion of accepted pseudo-samples. Or more exactly by the sum of the SMC weights at the end of the annealing simulation. This actually raises several questions: (a) this estimator is always between 0 and 1, while the marginal likelihood is not restricted [but this is due to a missing 1/ε in the likelihood estimate that cancels from both numerator and denominator]; (b) seeing the kernel as a non-parametric estimate of the likelihood led me to wonder why different ε could not be used in different models, in that the pseudo-data used for each model under comparison differs. If we were in a genuine non-parametric setting the bandwidth would be derived from the pseudo-data.

“Thus, Bayesian model selection by population annealing is valid.”

The discussion about the use of ABC population annealing somewhat misses the point of using ABC, which is to approximate the genuine posterior distribution, to wit the above quote: that the ABC Bayes factors favour the correct model in the simulation does not tell anything about the degree of approximation wrt the original Bayes factor. [The issue of non-consistent Bayes factors does not apply here as there is no summary statistic applied to the few observations in the data.] Further, the magnitude of the variability of the values of this Bayes factor as ε varies, from 1.3 to 9.6, mostly indicates that the numerical value is difficult to trust. (I also fail to explain the huge jump in Monte Carlo variability from 0.09 to 1.17 in Table 1.) That this form of ABC-SMC improves upon the basic ABC rejection approach is clear. However it needs to build some self-control to avoid arbitrary calibration steps and reduce the instability of the final estimates.

“The weighting function is set to be large value when the observed data and the simulated data are ‘‘close’’, small value when they are ‘‘distant’’, and constant when they are ‘‘equal’’.”

The above quote is somewhat surprising as the estimated likelihood f(x^{obs}|x^{obs},θ) is naturally constant when x^{obs}=x^{sim}… I also failed to understand how the model intervened in the indicator function used as a default ABC kernel

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