Multiplicative Factorization of Multi-Valued NIN-AND Tree Models

A multi-valued Non-Impeding Noisy-AND (NIN-AND) tree model has the linear complexity and is more expressive than common Causal Independence Models (CIMs). We formulate a Multiplicative Factorization (MF) for multi-valued NIN-AND Tree (NAT) models. In comparison with the MF for binary NAT models (of a undirected tree structure), the proposed MF is a hybrid and multiply connected graphical model. Although a NAT is made of two types of NIN-AND gates, we show that a sound and space efficient MF requires multiple types of gate MFs, and therefore significantly more sophisticated parameterization and integration of gate MFs, and soundness analysis. We show that the formulated MF is exact and its space complexity is linear on the number n of causes per effect. Based on the proposed MF, we extend the scheme for lazy propagation (LP) with binary NATmodeled Bayesian Networks (BNs) to multi-valued NATmodeled BNs. We show that the extended scheme is more powerful than LP based on MF of noisy-MAX. We demonstrate that the scheme allows significantly more efficient LP both in space and in time.