Application of tertiary structure of algebraic Bayesian network in the problem of a posteriori inference

In the theory of algebraic Bayesian networks, there are algorithms that allow to conduct a global posterior inference using secondary structures. At the same time, building secondary structures implies the use of tertiary structure. Consequently, the question about the separate application of the tertiary structure in the problem of a posterior inference arises. This issue has been considered earlier, but only a general description of the algorithm has been given, and only models with scalar estimates of the probability of truth have been taken into account. In this paper, we present an algorithm that extends the aforementioned algorithm to the possibility of using it in the case of interval estimates. In addition, an important property of an algebraic Bayesian network is acyclicality, and the correctness of the above-mentioned algorithms is ensured only for acyclic networks. Therefore, it is also necessary to be able to check the acyclicity of an algebraic Bayesian network using a tertiary structure. The description of this algorithm is also presented in this paper, it is based on the previously proved theorem that relates the number of knowledge pattern models in the network to the number of non-empty separators and the number of strong restriction connectivity components in acyclic algebraic Bayesian network, as well as the theorem proved in this paper that two knowledge pattern models belong to the same strong restriction connectivity component. For all the developed algorithms, the correctness of their performance is proved, and their time complexity estimation is calculated.