**A**nother arXived paper I surprisingly (?) missed, by George Papamakarios and Iain Murray, on an ABCDE (my acronym!) substitute to ABC for generative models. The paper was reviewed [with reviews made available!] and accepted by NIPS 2016. (Most obviously, I was *not* one of the reviewers!)

*“Conventional ABC algorithms such as the above suffer from three drawbacks. First, they only **represent the parameter posterior as a set of (possibly weighted or correlated) samples *[for which]* it is not obvious how to perform some other computations using samples, such as combining posteriors from two separate analyses. Second, the parameter samples do not come from the correct Bayesian posterior *(…)* Third, as the ε-tolerance is reduced, it can become impractical to simulate the model enough times to match the observed data even once *[when] *simulations are expensive to perform”*

The above criticisms are a wee bit overly harsh as, well…, Monte Carlo approximations remain a solution worth considering for all Bayesian purposes!, while the approximation [replacing the data with a ball] in ABC is replaced with an approximation of the true posterior as a mixture. Both requiring repeated [and likely expensive] simulations. The alternative is in iteratively simulating from pseudo-predictives towards learning better pseudo-posteriors, then used as new proposals at the next iteration modulo an importance sampling correction. The approximation to the posterior chosen therein is a mixture density network, namely a mixture distribution with parameters obtained as neural networks based on the simulated pseudo-observations. Which the authors claim [p.4] requires no tuning. (Still, there are several aspects to tune, from the number of components to the hyper-parameter λ [p.11, eqn (35)], to the structure of the neural network [20 tanh? 50 tanh?], to the number of iterations, to the amount of X checking. As usual in NIPS papers, it is difficult to assess how arbitrary the choices made in the experiments are. Unless one starts experimenting with the codes provided.) All in all, I find the paper nonetheless exciting enough (!) to now start a summer student project on it in Dauphine and hope to check the performances of ABCDE on different models, as well as comparing this ABC implementation with a synthetic likelihood version.

As an addendum, let me point out the very pertinent analysis of this paper by Dennis Prangle, 18 months ago!