Being Bayesian about Learning Bayesian Networks from Ordinal Data

In this paper we propose a Bayesian approach for inferring Bayesian network (BN) structures from ordinal data. Our approach can be seen as the Bayesian counterpart of a recently proposed frequentist approach, referred to as the ‘ordinal structure expectation maximisation’ (OSEM) method. Like for the OSEM method, the key idea is to assume that each ordinal variable originates from a Gaussian variable that can only be observed in discretized form, and that the dependencies in the latent Gaussian space can be modelled by BNs; i.e. by directed acyclic graphs (DAGs). Our Bayesian method combines the ‘structure MCMC sampler’ for DAG posterior sampling, a slightly modified version of the ‘Bayesian metric for Gaussian networks having score equivalence’ (BGe score), the concept of the ‘extended rank likelihood’, and a recently proposed algorithm for posterior sampling the parameters of Gaussian BNs. In simulation studies we compare the new Bayesian approach and the OSEM method in terms of the network reconstruction accuracy. The empirical results show that the new Bayesian approach leads to significantly improved network reconstruction accuracies.