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.