another neat example of how assumptions about noise and functional forms can let you do causal inference is in an exercise in "Elements of Causal Inference":
consider a linear model. The true model is Y ~ aX + ϵ, X causes Y. you want to distinguish, using observational data, from the case where Y causes X.
if the noise ϵ is Gaussian, there's no way to do this: there are reasonable models going both directions.
if you assume ϵ is uniformly distributed on some interval instead, then it becomes really obvious which way is the correct way.
the exercise recommends drawing little pictures with error bars to convince yourself of this, which is worth doing.
consider a linear model. The true model is Y ~ aX + ϵ, X causes Y. you want to distinguish, using observational data, from the case where Y causes X.
if the noise ϵ is Gaussian, there's no way to do this: there are reasonable models going both directions.
if you assume ϵ is uniformly distributed on some interval instead, then it becomes really obvious which way is the correct way.
the exercise recommends drawing little pictures with error bars to convince yourself of this, which is worth doing.