Optimization Monte Carlo: Efficient and Embarrassingly Parallel Likelihood-Free Inference
We describe an embarrassingly parallel, anytime Monte Carlo method for likelihood-free models. The algorithm starts with the view that the stochasticity of the pseudo-samples generated by the simulator can be controlled externally by a vector of random numbers u, in such a way that the outcome, knowing u, is deterministic. For each instantiation of u we run an optimization procedure to minimize the distance between summary statistics of the simulator and the data. After reweighing these samples using the prior and the Jacobian (accounting for the change of volume in transforming from the space of summary statistics to the space of parameters) we show that this weighted ensemble represents a Monte Carlo estimate of the posterior distribution. The procedure can be run embarrassingly parallel (each node handling one sample) and anytime (by allocating resources to the worst performing sample). The procedure is validated on six experiments.
Likelihood-free inference refers to methods used when it is expensive or impossible to evaluate the likelihood function and which instead perform inference by simulating data from the model. Existing methods, such as ABC, simulate data from many parameter values and accept only close matches to the observed data. This is wasteful as most simulations are typically poor matches. This paper proposes Optimisation Monte Carlo (OMC), a more efficient approach which views the simulator as a function f(,u) where  is the unknown parameters and u is a vector of U(0,1) draws. The idea is perform many iterations of the following: draw a u vector, minimise ||y-f(,u)|| (distance between observations and simulator output) to find the best , and weight it according to the the volume of nearby  values providing reasonably good fits. This weight is shown to depend on the Jacobian of f.OMC is an appealing novel and simple strategy. The paper is clearly written, original and seems technically correct. There are also several examples in which OMC convincingly beats state-of-the-art alternatives. However I have some doubts about significance: the examples are for fairly simple models and I am not convinced the approach will succeed for more complicated scenarios. In particular, I think in all the examples f(,u) is a smooth function of . If the simulator involves branching decisions affected by  this will not necessarily be the case and optimisation could become a lot more difficult, as it may be necessary to search every branch.A few other points are as follows:* I'd be interested to know how expensive a typical optimisation step is and whether it affects the runtime per sample of OMC compared to ABC methods. * For some models I imagine only a subset of u space is interesting and it would be nice to have a method which learns and concentrates on this. This is not really a criticism of OMC, as ABC methods don't do this either. * It might be interesting to comment on links with "Efficient likelihood-free Bayesian Computation for household epidemics" by Peter Neal, which seems to share some of the ideas used in OMC.Finally some possible typos:* Pg 3, "volume proportional" -> "volume is proportional" * Pg 4 equation (8), is hat needed on first f? * Pg 5, "this problem is" -> "this problem has" * Pg 6, "is simply the" -> "is simply"