An Algorithm for Learning the Essential Graph

Abstract

This article presents an algorithm for learning the essential graph of a Bayesian network. The basis of the algorithm is the Maximum Minimum Parents and Children algorithm developed by previous authors, with three substantial modifications. The MMPC algorithm is the first stage of the Maximum Minimum Hill Climbing algorithm for learning the directed acyclic graph of a Bayesian network, introduced by previous authors. The MMHC algorithm runs in two phases; firstly, the MMPC algorithm to locate the skeleton and secondly an edge orientation phase. The computationally expensive part is the edge orientation phase. The first modification introduced to the MMPC algorithm, which requires little additional computational cost, is to obtain the immoralities and hence the essential graph. This renders the edge orientation phase, the computationally expensive part, unnecessary, since the entire Markov structure that can be derived from data is present in the essential graph. Secondly, the MMPC algorithm can accept independence statements that are logically inconsistent with those rejected, since with tests for independence, a `do not reject' conclusion for a particular independence statement is taken as `accept' independence. An example is given to illustrate this and a modification is suggested to ensure that the conditional independence statements are logically consistent. Thirdly, the MMHC algorithm makes an assumption of faithfulness. An example of a data set is given that does not satisfy this assumption and a modification is suggested to deal with some situations where the assumption is not satisfied. The example in question also illustrates problems with the `faithfulness' assumption that cannot be tackled by this modification.