On the Completeness and Complexity of the Lifted Dynamic Junction Tree Algorithm

TL;DR

This paper contributes the first completeness and complexity analysis for a temporal lifted algorithm, the socalled lifted dynamic junction tree algorithm (LDJT), which is the only exact lifted temporal inference algorithm out there.

Abstract

For static lifted inference algorithms, completeness, i.e., domain liftability, is extensively studied. However, so far no domain liftability results for temporal lifted inference algorithms exist. In this paper, we close this gap. More precisely, we contribute the first completeness and complexity analysis for a temporal lifted algorithm, the socalled lifted dynamic junction tree algorithm (LDJT), which is the only exact lifted temporal inference algorithm out there. To handle temporal aspects efficiently, LDJT uses conditional independences to proceed in time, leading to restrictions w.r.t. elimination orders. We show that these restrictions influence the domain liftability results and show that one particular case while proceeding in time, has to be excluded from FO12 . Additionally, for the complexity of LDJT, we prove that the lifted width is in even more cases smaller than the corresponding treewidth in comparison to static inference.

Figures (14)

• Figure 1: Parfactor graph for $G^{ex}$
• Figure 2: $G_\rightarrow^{ex}$ the two-slice temporal parfactor graph for model $G^{ex}$
• Figure 3: Groundings due to Temporal Elimination Order ($J_{t-1}$ on the left and $J_t$ on the right)
• Figure 4: Lifted Mulltiplication. The definition assumes, without loss of generality that the logvars in the parfactors are standardized apart, i.e., the two parfactors do not share variable names (this can be achieved by renaming logvars).
• Figure 5: Lifted Summing-out.

Theorems & Definitions (38)

• Definition 1
• Definition 2
• Definition 3
• Example 1: Groundings LDJT cannot prevent
• Corollary 1
• proof
• Theorem 2
• proof
• Conjecture 1
• Theorem 3