# 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.

• Is (cross-validated) out of sample prediction accuracy an option?
– Eoin
Commented Oct 21, 2020 at 13:34
• i haven't had a chance to read this article, but this might be relevant.. ill take a look in a few hours. pnas.org/content/108/37/15112
– fool
Commented Oct 22, 2020 at 0:36
• @user228809 Thank you for the link. I've only read the abstract so far, but that seems to indicate that this is just not possible, or at least not well-founded due to unknown information loss. What a pitty... Commented Oct 22, 2020 at 7:29
• @Eoinisonthejobmarket I was also thinking about that, but I am not sure how that would work in this case. I am not sure if just drawing from the (unknown) distribution would be sufficient as a prediction, or if I have to add something. Also, I might not have enough data to do cross-validation. Commented Oct 22, 2020 at 7:31
• You said you can simulate data though? Isn't that a prediction? Also, if the data is small (and model fitting doesn't take too long), leave-one-out cross validation maybe?
– Eoin
Commented Oct 22, 2020 at 8:04