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Lifting Factor Graphs with Some Unknown Factors

Malte Luttermann, Ralf Möller, Marcel Gehrke

TL;DR

This work tackles probabilistic inference in factor graphs that include unknown factor potentials. It introduces the LIFAGU algorithm, a generalisation of the Colour Passing approach, which detects symmetric subgraphs that may include unknown factors and transfers potentials from known factors to these unknowns to obtain a well-defined semantics. The method relies on analyzing symmetric neighbourhoods and uses a threshold to control potential transfer, yielding a lifted representation comprised solely of known factors when the symmetry conditions hold. Empirical results show near-zero divergence between query distributions on fully known graphs and lifg-imputed graphs, while lifted inference with LVE outperforms standard VE in speed, demonstrating practical exactness and efficiency gains. Overall, LIFAGU provides a principled framework for extending lifted representations to partially known models, enabling scalable probabilistic reasoning under symmetry assumptions.

Abstract

Lifting exploits symmetries in probabilistic graphical models by using a representative for indistinguishable objects, allowing to carry out query answering more efficiently while maintaining exact answers. In this paper, we investigate how lifting enables us to perform probabilistic inference for factor graphs containing factors whose potentials are unknown. We introduce the Lifting Factor Graphs with Some Unknown Factors (LIFAGU) algorithm to identify symmetric subgraphs in a factor graph containing unknown factors, thereby enabling the transfer of known potentials to unknown potentials to ensure a well-defined semantics and allow for (lifted) probabilistic inference.

Lifting Factor Graphs with Some Unknown Factors

TL;DR

This work tackles probabilistic inference in factor graphs that include unknown factor potentials. It introduces the LIFAGU algorithm, a generalisation of the Colour Passing approach, which detects symmetric subgraphs that may include unknown factors and transfers potentials from known factors to these unknowns to obtain a well-defined semantics. The method relies on analyzing symmetric neighbourhoods and uses a threshold to control potential transfer, yielding a lifted representation comprised solely of known factors when the symmetry conditions hold. Empirical results show near-zero divergence between query distributions on fully known graphs and lifg-imputed graphs, while lifted inference with LVE outperforms standard VE in speed, demonstrating practical exactness and efficiency gains. Overall, LIFAGU provides a principled framework for extending lifted representations to partially known models, enabling scalable probabilistic reasoning under symmetry assumptions.

Abstract

Lifting exploits symmetries in probabilistic graphical models by using a representative for indistinguishable objects, allowing to carry out query answering more efficiently while maintaining exact answers. In this paper, we investigate how lifting enables us to perform probabilistic inference for factor graphs containing factors whose potentials are unknown. We introduce the Lifting Factor Graphs with Some Unknown Factors (LIFAGU) algorithm to identify symmetric subgraphs in a factor graph containing unknown factors, thereby enabling the transfer of known potentials to unknown potentials to ensure a well-defined semantics and allow for (lifted) probabilistic inference.
Paper Structure (7 sections, 2 theorems, 1 equation, 4 figures, 1 algorithm)

This paper contains 7 sections, 2 theorems, 1 equation, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Factor Graphs and Parameterised Factor Graphs
  4. The Colour Passing Algorithm
  5. The lifg Algorithm
  6. Empirical Evaluation
  7. Conclusion

Key Result

corollary thmcountercorollary

Given that for every unknown factor $f_i$ there is at least one known factor that is possibly identical to $f_i$ in an fg $G$, lifg is able to replace all unknown potentials in $G$ by known potentials.

Figures (4)

  • Figure 1: An fg for an epidemic example Hoffmann2022a with two individuals $alice$ and $bob$. The input-output pairs of the factors are omitted for simplification.
  • Figure 2: A pfg corresponding to the lifted representation of the fg depicted in \ref{['fig:epid_fg']}.The input-output pairs of the parfactors are again omitted for brevity.
  • Figure 3: The colour passing procedure of the cp algorithm on an exemplary input fg containing three Boolean rv without evidence and two factors with identical potentials. The example has been introduced by Ahmadi et al. Ahmadi2013a.
  • Figure 4: Left: The mean kl divergence on the queried probability distributions (thick line) as well as the standard deviation of all measured kl divergences for each choice of $d$ (ribbon around the mean). Right: The mean run time of ve and lve for each choice of $d$.

Theorems & Definitions (7)

  • definition thmcounterdefinition: Logvar, PRV, Event
  • definition thmcounterdefinition: Parfactor, Model, Semantics
  • definition thmcounterdefinition: 2-Step Neighbourhood
  • definition thmcounterdefinition: Symmetric 2-Step Neighbourhoods
  • definition thmcounterdefinition: Possibly Identical Factors
  • corollary thmcountercorollary
  • corollary thmcountercorollary