MATRIX-VECTOR ALGORITHMS FOR NORMALIZING FACTORS IN ALGEBRAIC BAYESIAN NETWORKS LOCAL POSTERIORI INFERENCE

Abstract

We consider a task of local posteriori inference description by means of matrix-vector equations in algebraical Bayesian networks that represent a class of probabilistic graphical models. Such equations were generally presented in previous publications, however containing normalizing factors that were provided with algorithmic descriptions of their calculations instead of the desired matrix-vector interpretation. To eliminate this gap, the normalized factors were firstly represented as scalar products. Then, it was successfully shown that one of the components in each scalar product can be expressed as a Kronecker degree of a constant two-dimensional vector. Later on, non-normalized posteriori inference matrixoperator transplantation and further transfer within each scalar product yielded a representation of one of the scalar product components as a sequence of tensor products of two-dimensional vectors. The latter vectors have only two possible values in one case and three values in the other. The choice among those values is determined by the structure of input evidence. The second component of each scalar products is the vector with original data. The calculations performed gave the possibility for constructing corresponding vectors; the paper contains a table with proper examples for some of them. Local posteriori inference representation for matrix-vector equations simplify the development of local posteriori inference algorithms, their verification and further implementation based on available libraries. These equations also give the possibility for application of classical mathematical techniques to the obtained results analysis. Finally, the results obtained make it possible to apply the method of postponed calculations. This method helps avoiding construction of big-size vectors; instead, the vectors components can be calculated just in time they are needed by means of bitwise operations.

Keywords

• Bayesian networks
• posteriori inference
• inference algorithms
• postponed calculations
• bitwise operations