I read a post explaining why the Akaike Criterion cannot be used for deciding if A cause B or B caused A.
I'm curious about a more general case of using AIC for causal inference (with observational data only). Consider the case of 3 different events labelled A,B,C and we have a non-trivial joint probability distribution over them. For sanity, let's assume we have a partial order ($C<A$ and $C<B$). So the potential causal structures (excluding the possibility of latent variables) are C causes A, C causes A and C causes B, C causes B, A causes B, B causes A, C causes A and C causes B and A causes B, etc.
For each causal structure we could assign a parameterized causal model, the parameters determining the dependencies and initial distribution for any exogenous variables. Ex. if A,B all take on N values and C takes on M, the causal model for the "C causes A" case would have M*(N-1)+(M-1)+(N-1) parameters.
Then you assume independent normally distributed errors in your observed probabilities. Restrict yourself to one causal structure and find the parameters to minimize the Chi-squared. You then compare the minimums of each structure using an parameter number penalty like AIC.
Ex. let's say the lowest Chi-squared value for the "C causes A" case is 150 and lowest for "C cause A and C causes B" is 140 but the second model is more complex so when the parameter penalizations are accounted for it has a worse score.
Could an AIC criterion of sorts work for the problem posed in this way?
I recognize in some cases there will be multiple structures that could give rise to the same joint probability distribution and therefore potentially the same minimum Chi-squared score however in those cases the AIC criterion would select the least complicated structure or if both have the same complexity it cannot distinguish these two and ranks them equally. This I believe is the issue raised when thinking about if A causes B or B causes A since any joint distribution can be obtained either way and they have same complexity but this would not be the case for two general causal models.