Respecting Markov equivalence in computing posterior probabilities of causal graphical features

There have been many efforts to identify causal graphical features such as directed edges between random variables from observational data. Recently, Tian et al. proposed a new dynamic programming algorithm which computes marginalized posterior probabilities of directed edge features over all the possible structures in O(n3ⁿ) time when the number of parents per node is bounded by a constant, where n is the number of variables of interest. However the main drawback of this approach is that deciding a single appropriate threshold for the existence of the directed edge feature is difficult due to the scale difference of the posterior probabilities between the directed edges forming v-structures and the directed edges not forming v-structures. We claim that computing posterior probabilities of both adjacencies and v-structures is necessary and more effective for discovering causal graphical features, since it allows us to find a single appropriate decision threshold for the existence of the feature that we are testing. For efficient computation, we provide a novel dynamic programming algorithm which computes the posterior probabilities of all of n(n-1)/2 adjacency and n(₂^n-1) v-structure features in O(n³3ⁿ) time.