Complexity measurement of fundamental pseudo-independent models

Pseudo-independent (PI) models are a special class of probabilistic domain model (PDM) where a set of marginally independent domain variables shows collective dependency, a special type of dependency associated with the scope of a set of variables in a probabilistic domain. Due to this property, common probabilistic learning methods based on a single-link lookahead search cannot learn PI models. To learn PI models, a learning algorithm should be equipped with a search with its scope beyond a single link, which is called a multi-link lookahead search. An improved result can be obtained by incorporating model complexity into a scoring metric to explicitly trade off model accuracy for complexity and vice versa during selection of the best model among candidates at each learning step. To implement this scoring metric for learning PI models, the complexity formula for every class of PI models is required. Previous studies found the complexity formula for full PI models, one of the three major types of PI models (the other two are partial and mixed PI models). This study presents the complexity formula for atomic partial PI models, partial PI models that contain no embedded PI submodels. This paper shows the complexity can be acquired by arithmetic operation with the cardinality of the space of domain variables in an atomic partial PI model. The new formula provides the basis for further characterizing the complexity of non-atomic PI models, which contain embedded PI submodels in their domains. 2006 Elsevier Inc. All rights reserved.