Model comparison with intractable likelihood using approximate Bayesian Computation I have some models based on stochastic differential equations (SDEs). Because of the definition of these models, I can simulate data, but I cannot compute the likelihood function / distribution function. Therefore, I currently plan to use approximate Bayesian computation (ABC) to fit the parameters of these models.
However, I also need a method to compare different SDEs, which are currently discussed as possible explanations of the data, while accounting for the complexity of the parameters. Normally, I would compare these models based on DIC, LOOIC etc, but all these require the likelihood to be known.
Is there any method for comparing the model complexity, if the likelihood is unknown?
The only way I could think that might work, is to use a Bayesian model selection (i.e. using a categorial variable to switch between the models), but I am not sure if this would work at all.
 A: There are examples in the ABC literature of model selection through Bayes factors. An ecological individual based model example is here:
https://doi.org/10.1016/j.ecolmodel.2017.07.017
The paper involves picking between models of different complexity so hopefully it will be useful even if it doesn't deal with SDEs directly.
However, I should also mention that there are those that think that Bayes factors (or odds ratios or any other kind of Bayesian model selection) is just philosophically unsound. This is an argument for example forcefully stated by Gelman and Shalizi.
http://doi.org/10.1111/j.2044-8317.2011.02037.x
Their argument is that models ought to be judged primarily by their simulation output. Rather than a formula then you ought to come up with a series of model checks where you compare simulation outputs with either data you held out or other features of the data that weren't directly part of the fitting (in your case, since it's ABC, that would mean other summary statistics that were not included in the rejection step).
In my view this gives a much better grip at what the model is good and bad and is a lot more convincing than citing some obscure probability measure.  Of course though, this is a very context-dependent exercise.
