Computationally Efficient Methods For Selecting Among Mixtures Of Graphical Models, With Discussion
We describe computationally efficient methods for Bayesian model selection. The methods select among mixtures in which each component is a directed acyclic graphical model (mixtures of DAGs or MDAGs), and can be applied to data sets in which some of the random variables are not always observed. The model-selection criterion that we consider is the posterior probability of the model (structure) given data. Our model-selection problem is difficult because (1) the number of possible model structures grows super-exponentially with the number of random variables and (2) missing data necessitates the use of computationally slow approximations of model posterior probability. We argue that simple search-and-score algorithms are infeasible for a variety of problems, and introduce a feasible approach in which parameter and structure search is interleaved and expected data is treated as real data. Our approach can be viewed as a combination of the Cheeseman-Stutz asymptotic approximation for model posterior probability and the Expectation-Maximization algorithm. We evaluate our procedure for selecting among MDAGs on synthetic and real examples.