Causal versions of maximum entropy and principle of insufficient reason

The principle of insufficient reason (PIR) assigns equal probabilities to each alternative of a random experiment whenever there is no reason to prefer one over the other. The maximum entropy principle (MaxEnt) generalizes PIR to the case where statistical information like expectations are given. It is known that both principles result in paradoxical probability updates for joint distributions of cause and effect. This is because constraints on the conditional P ( effect ∣ cause ) P\left({\rm{effect}}| {\rm{cause}}) result in changes of P ( cause ) P\left({\rm{cause}}) that assign higher probability to those values of the cause that offer more options for the effect, suggesting “intentional behavior.” Earlier work therefore suggested sequentially maximizing (conditional) entropy according to the causal order, but without further justification apart from plausibility on toy examples. We justify causal modifications of PIR and MaxEnt by separating constraints into restrictions for the cause and restrictions for the mechanism that generates the effect from the cause. We further sketch why causal PIR also entails “Information Geometric Causal Inference.” We briefly discuss problems of generalizing the causal version of MaxEnt to arbitrary causal DAGs.