An extended view on lifting Gaussian Bayesian networks

Lifting probabilistic graphical models and developing lifted inference algorithms aim to use higher level groups of random variables instead of individual instances. In the past, many inference algorithms for discrete probabilistic graphical models have been lifted. Lifting continuous probabilistic graphical models has played a minor role. Since many real-world applications involve continuous random variables, this article turns its focus to lifting approaches for Gaussian Bayesian networks. Specifically, we present algorithms for constructing a lifted joint distribution for scenarios of sequences of overlapping and non-overlapping logical variables. We present operations that work in a fully lifted way including addition, multiplication, and inversion. We present how the operations can be used for lifted query answering algorithms and extend the existing query answering algorithm by a new way of evidence handling. The new way of evidence handling groups evidence that has the same effect on its neighboring variables in cases of partial overlap between the logical-variable sequences. In the theoretical complexity analysis and the experimental evaluation, we show under which conditions the existing lifted approach and the new lifted approach including evidence grouping lead to the most time savings compared to the grounded approach.