This chapter discusses Bayes Factors as they apply to Chain Event Graph (CEG) model selection. It presents a recent method based on dynamic programming which is able to systematically and efficiently search the full space of stratified CEGs to find the maximum a posteriori (MAP) CEG in this class strategies. The chapter discusses the exhaustive and approximative model search algorithms developed for searching over a CEG model space. It describes the agglomerative hierarchical clustering algorithms presented which explore only local neighbourhoods of already found models to the more general approach of exhaustively searching a model space using dynamic programming (DP) techniques. The chapter shows that local priors (LPs) are also prone to select more complex models when Bayes Factor (BF) is used for CEG model selection. It outlines some properties of the pairwise moment non-local priors (NLPs) which constitute a new class of NLPs specifically customised for discrete processes developing over trees.