Probabilistic relational models such as parametric factor graphs enable efficient (lifted) inference by exploiting the indistinguishability of objects. In lifted inference, a representative of indistinguishable objects is used for computations. To obtain a relational (i.e., lifted) representation, the Advanced Colour Passing (ACP) algorithm is the state of the art. The ACP algorithm, however, requires underlying distributions, encoded as potential-based factorisations, to exactly match to identify and exploit indistinguishabilities. Hence, ACP is unsuitable for practical applications where potentials learned from data inevitably deviate even if associated objects are indistinguishable. To mitigate this problem, we introduce the ε-Advanced Colour Passing (ε-ACP) algorithm, which allows for a deviation of potentials depending on a hyperparameter ε. ε-ACP efficiently uncovers and exploits indistinguishabilities that are not exact. We prove that the approximation error induced by ε-ACP is strictly bounded and our experiments show that the approximation error is close to zero in practice.

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.

Gaussian Bayesian networks are widely used for modeling the behavior of continuous random variables. Lifting exploits symmetries when dealing with large numbers of isomorphic random variables. It provides a more compact representation for more efficient query answering by encoding the symmetries using logical variables. This paper improves on an existing lifted representation of the joint distribution represented by a Gaussian Bayesian network (lifted joint), allowing overlaps between the logical variables. Handling overlaps without grounding a model is critical for modelling real-world scenarios. Specifically, this paper contributes (i) a lifted joint that allows overlaps in logical variables and (ii) a lifted query answering algorithm using the lifted joint. Complexity analyses and experimental results show that - despite overlaps - constructing a lifted joint and answering queries on the lifted joint outperform their grounded counterparts significantly.