Extended BIC Criterion for Model Selection

Model selection is commonly based on some variation of the BIC or minimum message length criteria, such as MML and MDL. In either case the criterion is split into two terms: one for the model (data code length/model complexity) and one for the data given the model (message length/data likelihood). For problems such as change detection, unsupervised segmentation or data clustering it is common practice for the model term to comprise only a sum of sub-model terms. In this paper it is shown that the full model complexity must also take into account the number of sub models and the labels which assign data to each sub model. From this analysis we derive an extended BIC approach (EBIC) for this class of problem. Results with artificial data are given to illustrate the properties of this procedure.