Quaternary structure synthesis in algebraic bayesian networks: incremental and decremental algorithms

Abstract

Subject of Research. Algebraic Bayesian networks are referred to a class of probabilistic graphical models that are a representation of knowledge bases with uncertainty. The distinguishing feature of ABN is the availability of global structures. Among them there are primary and secondary structures that are directly used in various kinds of probabilistic logical inference as well as tertiary and quaternary involved in the problems of automatic synthesis and identification of the properties of the secondary structure and partially in the machine learning tasks within specified networks. Existing algorithms for quaternary structure changing require its complete rebuild when changing the primary structure. That feature slows down the whole global structures synthesis, dispels user’s attention who is forced to re-analyze the whole rebuilt structure instead of focusing on the changes that were directly caused by the limited modification of the original data. This fact reduces ABN attractiveness as a model for data processing in general. Scope of Research. This paper is aimed at speeding up the rebuild process and eliminating the shortage of the quaternary structure rebuild algorithms when adding and deleting vertices of primary structure expressed in excess rebuild of the entire structure. The task of algorithm incrementalization for quaternary structure rebuild is solved to achieve the goal. Method. The proposed approach is based on the properties of incremental algorithms that reduce the amount of computations due to the result obtained at the previous step of the algorithm. All the arguments used in the paper are expressed in a graph theory language to apply the established system of terms and classical results. Main Results. The paper presents incremental and decremental algorithms, complemented by a proof of correctness and listing. Given algorithms are based on the previously obtained incremental algorithms for tertiary structure. Moreover, a detailed analysis of the plurality of separators is carried out on each stage of the algorithms. Theoretical and Practical Relevance. These algorithms develop a global structure of algebraic Bayesian networks as well as the theory of probabilistic graphical models in general. Furthermore, they create the groundwork for creation of the secondary structure invariants that may be non-unique even if the primary structure is fixed unlike the current approach where connections are included in the secondary structure. The deletion of manipulation with a set of secondary structures considerably simplifies and makes foreseeable visualization of such complex object as ABN and may improve the computational characteristics of probabilistic logical inference algorithms. It also enables to reformulate the problem of ABN machine learning excluding any need for synthesis of many different objects with complex structure and virtually the same semantics. It also can be expected that the obtained incremental algorithms will accelerate computational processes of rebuilding and properties analysis for all four global ABN structures.