A Bayesian belief network directed cycle with multinomial random variables

The paper generalizes the transformation of a directed cycle in Bayesian belief networks (BBN) with binary random variables into a knowledge patterns chain in algebraical Bayesian networks (ABN) for the case of multivariate random variables. Under the assumption that multivariate random variables are represented with binary random variables conjuncts, the generalized transformation consists of the same steps as the original one. First, we form stochastic matrices that correspond to conditional probability tensors in the cycle nodes. Then we calculate the product of the matrices and find out the stochastic eigen-vector of the product result. The eigen-vector represents the probabilistic distribution of cycle node random variable assignments. Later on, this distribution is used in calculations of joint distributions for random variables assignments in couples of neighboring nodes. Finally, an ABN knowledge patterns cycle is constructed with the set of latter joint distributions, and then an ABN knowledge pattern chain is constructed with the latter cycle. The method for the chain reconciliation is known. Keywords: directed cycle, Bayesian network, random variable.