Lifting Factor Graphs with Some Unknown Factors for New Individuals

TL;DR

The paper addresses the challenge of probabilistic inference in factor graphs that include unknown factor mappings, proposing lifg, a generalisation of the Advanced Colour Passing (ACP) algorithm, to produce a lifted parameterised factor graph (pfg) $G'$ by transferring known potentials to unknown factors. lifg leverages indistinguishable 2-step neighbourhoods and a threshold $\theta$ to identify possibly identical factors $f_i \approx f_j$ and to group unknown factors with compatible known factors, thereby preserving well-defined semantics. It further extends lifg with background knowledge that specifies which factors belong to the same individual, enabling more informed and fewer ambiguous transfers. An empirical evaluation demonstrates that lifg yields query results closely matching the ground truth (low KL-divergence) and provides practical speedups over traditional inference, validating its effectiveness under missing-potential scenarios and highlighting its potential for real-world incomplete-data settings.

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 unknown factors, i.e., factors whose underlying function of potential mappings is unknown. We present the Lifting Factor Graphs with Some Unknown Factors (LIFAGU) algorithm to identify indistinguishable subgraphs in a factor graph containing unknown factors, thereby enabling the transfer of known potentials to unknown potentials to ensure a well-defined semantics of the model and allow for (lifted) probabilistic inference. We further extend LIFAGU to incorporate additional background knowledge about groups of factors belonging to the same individual object. By incorporating such background knowledge, LIFAGU is able to further reduce the ambiguity of possible transfers of known potentials to unknown potentials.

Figures (9)

• Figure 1: An fg for an epidemic example Hoffmann2022a with two individuals $alice$ and $bob$ as well as two medications $m_1$ and $m_2$ for treatment. For simplicity, all rv are Boolean and the input-output pairs of the factors are omitted.
• Figure 2: A pfg corresponding to the lifted representation of the fg depicted in \ref{['fig:epid_fg']}. The input-output pairs of the pf are again omitted for brevity.
• Figure 3: The colour passing procedure of the acp algorithm on an exemplary input fg Ahmadi2013a containing three Boolean rv without evidence and two factors encoding functions with identical potential mappings.
• Figure 4: An extension of the epidemic example depicted in \ref{['fig:epid_fg']}. The factors $f_{?}$ are unknown. The input-output pairs of the remaining factors are again omitted for brevity.
• Figure 5: An illustration of the steps undertaken by \ref{['alg:lifg']} (lifg) on the input fg depicted in \ref{['fig:epid_fg_extended_01']}: (a) Colouring of known factors (\ref{['line:assign_known_colours']}), (b) initial colouring of unknown factors (\ref{['line:assign_unknown_colours']}), (c) refined colouring for unknown factors (\ref{['line:reassign_unknown_colours']}), and (d) grouping of unknown factors with known factors (\ref{['line:assign_unkown_from_known', 'line:transfer_unkown_from_known']}).

Theorems & Definitions (25)

• Definition 2.1: Factor Graph, Kschischang2001a
• Example 2.1
• Definition 2.2: Parameterised Random Variable
• Example 2.2
• Definition 2.3: Parfactor
• Example 2.3
• Definition 2.4: Parametric Factor Graph
• Example 2.4
• Definition 2.5: Unknown Factor
• Example 2.5