Static Analytical Reasoning of Directed Cyclic Graph in the Discrete Case Weight Combination Method

As being defined, a Bayesian Network (BN) is based on a directed acyclic graph (DAG). However, when statistical data to learn a BN are collected in a set of groups independently by different experts, in other words, when we do not have the entire data covering all variables, we may have to construct a set of small pieces of BNs independently based on these different groups of data, and then combine these BNs together as a whole BN by fusing the same variables in different BNs. But directed cyclic graphs (DCGs) may appear. If we allow such cyclic BN, what is its definition? What is its physical meaning? What is its algorithm to make the inference? The answers are provided in this paper. Note that a cyclic BN cannot be understood as a dynamic BN (DBN), because variables are not in a doubtless time sequence.