Compression versus Accuracy: A Hierarchy of Lifted Models

Jan Speller; Malte Luttermann; Marcel Gehrke; Tanya Braun

REVIEW 2 major objections 5 minor 24 references

Compression versus Accuracy: A Hierarchy of Lifted Models

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that a precomputed hierarchy of tolerance values makes approximate lifted model construction hyperparameter-free while preserving the sharp worst-case accuracy bounds of ε-ACP.

desk verdict Hierarchy construction is sound and useful, but the inherited accuracy bound is false on a simple two-factor example. read the letter →

arxiv 2505.22288 v2 pith:M5XMO3SK submitted 2025-05-28 cs.AI

Jan Speller , Malte Luttermann , Marcel Gehrke , Tanya Braun This is my paper

classification cs.AI

keywords liftedinferenceprobabilisticgraphicalmodelsfactorgraphsepsilon-equivalencehierarchicalmodelconstructioncolourpassingapproximateerrorbounds

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Probabilistic graphical models with many interchangeable objects can be compressed into smaller lifted models by grouping factors whose potential tables are nearly equal. The established ε-ACP algorithm performs this grouping with a user-chosen tolerance ε, but each ε requires a separate run and different ε values can produce incompatible groupings. This paper proposes hierarchical ACP (HACP), which precomputes all tolerance values at which the grouping should change, in an order that guarantees a nested hierarchy: any factors grouped at a low tolerance remain grouped at every higher tolerance. The paper claims that this hierarchical construction keeps the same worst-case query-deviation bounds as ε-ACP, so a user can choose a tolerance level after seeing the full compression-accuracy trade-off rather than searching for one. The point of the paper is to make approximate lifted model construction hyperparameter-free and interpretable while preserving formal error guarantees.

What carries the argument

The central object is 1DEED, the one-dimensional ε-equivalence distance $d_\infty(\phi_1,\phi_2)=\max_k |\phi_1(k)-\phi_2(k)|/\min(|\phi_1(k)|,|\phi_2(k)|)$, which condenses the pairwise ε-equivalence check into a single number: Theorem 2 states $\phi_1 =_\varepsilon \phi_2 \iff d_\infty(\phi_1,\phi_2)\le\varepsilon$. Algorithm 1 uses these scalar distances to perform complete-linkage agglomerative merging—each merge records the largest pairwise distance among all factors in the combined group, so the stored ε is the smallest value at which the whole group is pairwise ε-equivalent. Algorithm 2 then reads off the groups at any chosen level, assigns ACP colours, runs standard colour passing, and replaces each group's potentials by their mean. The error analysis is carried by the Chan-Darwiche distance $D_{\mathrm{CD}}(P_M,P_{M'})$, through which Theorem 3's sharp ε-ACP bound is transferred to HACP and Theorem 5 converts it into an absolute query-deviation bound $p_{\max\Delta}\le(\sqrt{e^d}-1)/(\sqrt{e^d}+1)$.

What would settle it

Compute $D_{\mathrm{CD}}(P_M,P_{M'})$ for the original factor graph and the HACP-compressed graph at a hierarchy level with tolerance ε, on the extremal example used to show Theorem 3 is sharp: if the distance exceeds the bound in Eq. (5), Proposition 4 is refuted; if the extremal grouping is unreachable under HACP's nesting constraint, the claim that the bound remains sharp for HACP has no witness.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that approximate lifted model construction can be made hierarchical without sacrificing its accuracy guarantee. It introduces the one-dimensional ε-equivalence distance $d_\infty(\phi_1,\phi_2)=\max_k |\phi_1(k)-\phi_2(k)|/\min(|\phi_1(k)|,|\phi_2(k)|)$, proves that two potential vectors are ε-equivalent exactly when this distance is at most ε, and uses the pairwise distances to build an agglomerative hierarchy in which merging two groups records their maximal pairwise distance. The resulting HACP algorithm colours each group and runs standard ACP, then replaces potentials by their group mean; because merges are only ever refined at coarser levels, the induced models form a strict nesting. The paper further claims that the sharp Chan-Darwiche error bound proved for ε-ACP applies unchanged to every level of the HACP hierarchy (Proposition 4, using Theorem 3), and derives a closed-form expression for the largest ε that keeps any query's absolute deviation below a target $p^*_\Delta$. All of this is meant to let a practitioner choose a compression level by examining the precomputed hierarchy and its error bounds, without guessing or exploring ε values.

Load-bearing premise

The accuracy guarantee depends on transferring the sharp ε-ACP worst-case bound to HACP even though HACP searches a strictly smaller space of allowed groupings, and the transfer is justified by a proof sketch that defers to the earlier bound's proof rather than re-deriving it.

Editorial extensions

If this is right

Running Algorithm 1 once on a factor graph yields every ε at which the grouping changes, so a user can pick a compression level after seeing the trade-off instead of running ε-ACP repeatedly.
The models at different levels are nested: a factor group formed at a smaller ε remains a subset of a group at every larger ε, making the models directly interpretable with respect to one another.
Every HACP level inherits the same sharp Chan-Darwiche deviation bound as ε-ACP, so the hierarchy does not weaken the formal accuracy guarantee.
Given a target maximum query deviation $p^*_\Delta$, Theorem 7 returns the largest ε that still meets it, turning an accuracy requirement into a concrete hierarchy level.
Increasing ε and increasing the number of factors $m$ affect the bound in similar ways (Note 2), giving a simple rule of thumb for how much compression can be bought at a given accuracy cost.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: Because the hierarchy is a complete-linkage dendrogram with recorded merge levels, the same precomputed ε vector can also serve as a diagnostic of symmetry structure—large gaps between consecutive ε values indicate genuine asymmetries between factor populations, while tight clusters of ε values indicate near-duplicate factors.
Editorial inference: A natural empirical extension is to measure actual query deviations of HACP against ε-ACP on standard relational factor graphs, since the paper proves only that the worst-case bound transfers and notes that HACP may show somewhat higher average deviations.
Editorial inference: The hierarchy could be used to recommend a default level automatically—for example, the smallest ε whose additional compression is negligible under the bound, or the largest ε before the bound exceeds a target deviation—without additional inference runs.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The hierarchy construction is a real contribution, but the central accuracy guarantee does not survive contact with a two-factor example. Under the paper's own definitions, Proposition 4 is false. Take two unary factors φ_A=(1,1+ε) and φ_B=(1+ε,1+ε). Their 1DEED is ε, so Algorithm 1 merges them at level ε and Algorithm 2 replaces both by the mean (1+ε/2,1+ε). The Chan–Darwiche distance between original and compressed joint distributions is 2 ln(1+ε) (largest ratio (1+ε/2)(1+ε) at A=T,B=F; smallest (1+ε/2)/(1+ε) at A=F,B=T). Eq. (5) with m=2 gives ln((1+ε)/(1+ε/2)), which is strictly smaller for every ε>0. So the claimed sharp bound is not a bound. The looser bounds in Eq. (6) still seem to hold, so a repair is plausible, but the paper's assertion of identical worst-case guarantees for HACP collapses.

What is good: the 1DEED characterization (Theorem 2) is clean and correctly proved; the agglomerative hierarchical construction with complete linkage is a sound way to enforce nested groupings and is genuinely hyperparameter-free; the O(m^3) pre-processing cost is reasonable. The paper also gives careful proofs for the absolute deviation bounds (Theorems 5 and 7). The interpretability discussion is sensible, though unvalidated.

Soft spots beyond Proposition 4: no experiments at all, so the practical compression/accuracy trade-off claims are untested. The proof of Proposition 4 is a sketch deferring to [14] without re-deriving the bound on the restricted hierarchical grouping space—and that deferral is exactly where the argument fails. The counterexample also suggests that Theorem 3 in [14] may itself be mis-stated; either way, this paper inherits the problem.

Who should read it: researchers working on approximate lifted inference who want a hyperparameter-free model hierarchy. The algorithmic idea is worth discussing, but the accuracy guarantee needs to be re-derived, not just transferred.

I would send it to peer review, because the hierarchy construction is novel and the flaw is concrete and fixable; a serious referee should catch it. With a corrected bound—even the looser one—the paper could be solid.

Referee Report

2 major / 5 minor

Summary. The paper proposes Hierarchical Advanced Colour Passing (HACP), a hyperparameter-free method for constructing a hierarchy of approximate lifted probabilistic graphical models. It introduces a one-dimensional ε-equivalence distance (1DEED) and proves in Theorem 2 that this distance exactly characterizes ε-equivalence. Algorithm 1 computes a nested hierarchy of factor groupings and an associated ordered vector of ε values using complete-linkage agglomerative clustering on the 1DEED matrix. Algorithm 2 converts any selected hierarchy level into a parametric factor graph by assigning group colors and replacing factor potentials by group means. The paper claims that HACP inherits the exact error bound of ε-ACP (Proposition 4) and derives further bounds on maximal query deviations (Theorem 5, Corollary 6, Theorem 7).

Significance. If the error-bound transfer is made rigorous, the paper fills a genuine need in approximate lifted inference: it removes the user-defined ε hyperparameter and yields nested, interpretable models. The 1DEED characterization (Theorem 2) and the probabilistic deviation bounds (Theorem 5) are proven in detail in the appendix. The main weaknesses are that Proposition 4 is only a proof sketch, and the practical claims about interpretability and efficiency are not validated empirically. The theoretical contribution is otherwise coherent and appears correct under a standard reading of the equations.

major comments (2)

[4.1, Proposition 4] The proof of Proposition 4 is a two-sentence sketch that defers to the original proof in [14, App. A] without stating or verifying the hypotheses under which Theorem 3's bound applies to HACP. Because HACP restricts the space of possible groupings relative to ε-ACP, the transfer is not automatic; the paper should either provide a full proof in the appendix or explicitly list the conditions (e.g., every group consists of pairwise ε-equivalent factors and potentials are replaced by the group mean) and verify them. The optimality claim—that the same example as in [14] hits the bound—is also asserted without checking that the tightness configuration is reachable under the hierarchical constraint. I examined the specific counterexample proposed in the review: for m=2, Eq. (5) reads ln(((1+ε/2)(1+ε))/(1+ε/2))^2 = ln((1+ε)^2) = 2 ln(1+ε), not ln((1+ε)/(1+ε/2)), so that example does not falsify the bound. The substantive gap is therefore the missing proof, not the counterexample.
[3.3, Algorithm 2, line 5] The condition `j < m` in the definition of GΦ(k) incorrectly excludes factor index m from any group. Since factor indices run from 1 to m, the condition should be `j ≤ m` (or equivalently `j < m+1`). As written, whenever factor m belongs to a non-singleton group at a given hierarchy level, it would be omitted from that group and later treated as a singleton, yielding an incorrect grouping. This is a concrete bug in a central algorithm and should be fixed in the pseudocode.

minor comments (5)

[4.1, Eq. (5) and Eq. (10)] The exponent m is typeset in a way that makes it easy to misread as applying only to the denominator. The proof of Theorem 7 (taking the m-th root of e^d) shows that the exponent applies to the entire fraction. Please typeset as `\left( \frac{(1+\frac{m-1}{m}\varepsilon)(1+\varepsilon)}{1+\frac{1}{m}\varepsilon} \right)^m` to remove ambiguity.
[3.2, Algorithm 1] When multiple entries of the distance matrix tie for the minimum, the resulting hierarchy and ε-vector are not unique. The paper does not discuss tie-breaking or its effect on the claimed nested structure and on the error bounds. A deterministic tie-breaking rule should be specified, or the non-uniqueness should be acknowledged.
[4.2, Theorem 7] The displayed formula for ε1 appears garbled: the fractions and square root are distorted, and the expression as presented is hard to parse. The derivation in the proof gives a valid quadratic, but the main-text formula should be corrected for reproducibility.
[5 (Discussion)] The paper makes practical claims about interpretability and efficiency gains (e.g., 'enables interpretability between the different models') but provides no experimental evaluation. A small case study on a standard benchmark would substantially strengthen these claims, even for a primarily theoretical paper.
[3.2, opening paragraph] The input specification 'Φ ⊂ R^{n×m}_{>0}' is a misuse of notation: Φ is a set of m factors, each a vector in R^n_{>0}, not an n×m matrix. The wording 'with m = |Φ| and Φ ⊂ R^{n×m}_{>0}' should be corrected to 'Φ = {ϕ_1,...,ϕ_m} with ϕ_i ∈ R^n_{>0}'.

Circularity Check

1 steps flagged · score 4.0 of 10

Proposition 4's error-bound transfer is a load-bearing self-citation; the hierarchy construction itself is independent.

self citation load bearing [Section 4.1, Proposition 4 (proof), relying on Theorem 3 from [14]; reinforced in Section 5]
"Theorem 3 (Luttermann et al. [14]) ... where the bound given in Eq. (5) is optimal (sharp). Proposition 4. Theorem 3 holds the same way for M′ being the output of the HACP algorithm (Alg. 2). Proof. The core components of the ε-ACP algorithm and its hierarchical counterpart HACP (Alg. 2) are identical, aside from enforcing predefined group structures to guarantee a hierarchical structure. Therefore, the proof can be conducted in the same manner as the original proof [14, App. A]. The same proposed example can be used to hit the bound of Eq. (5), showing its optimality."

Proposition 4 is the paper's accuracy guarantee, but it does not derive a HACP-specific bound. It transfers Theorem 3 from the authors' own prior work [14] by asserting the algorithms are identical 'aside from enforcing predefined group structures.' That aside is exactly the HACP difference: the forced complete-linkage hierarchy restricts reachable groupings, so the worst case and the 'same proposed example' used for sharpness must be re-verified in this restricted space. Instead, both the bound and its optimality are lifted from [14, App. A] via citation. The paper's later statement that both algorithms retain identical worst-case bounds 'by construction' thus rests on a self-citation rather than on a derivation, making the cited theorem load-bearing.

full rationale

The hierarchical construction itself is self-contained: Theorem 2 is an equivalent reformulation of Definition 3, Algorithm 1's nested merge rule guarantees the hierarchy property by construction, and Algorithm 2's mean-potential update is defined in-text. The single load-bearing circular dependency is the transfer of the sharp ε-ACP error bound to HACP in Proposition 4, whose proof defers entirely to the authors' own [14] and assumes the optimality example carries over without verifying that the HACP grouping restriction permits it. I do not score higher because the paper's main algorithmic contribution (the hierarchy of ε values and nested models) does not reduce to a fit or to a self-citation; the problem is localized to the inherited accuracy claim. The skeptic's two-factor counterexample, if correct, would make Proposition 4 false rather than merely cited; either way, it underscores that the bound's transfer is assumed, not derived.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the positive-potential assumption, the identity-permutation simplification, the structural compatibility requirement for grouping, and the cited epsilon-ACP error bound. No free parameters are fitted; the hierarchy epsilon values are outputs of the algorithm.

assumptions (5)

domain assumption All potentials are strictly positive (Def. 1).
Stated as a stipulation to avoid division by zero in d_infinity; zeros are said to be replaceable by tiny numbers. This is an assumption about the input models.
domain assumption Permutation pi in epsilon-equivalence is the identity (Sec. 2).
Stated for simplicity; the authors claim all results apply to any permutation pi [14], but the proofs are only given for identity.
domain assumption Factors to be grouped share identical dimension and range structure (structural compatibility, Alg. 1 input).
The hierarchy algorithm requires pairwise distances between all factors, and grouping only makes sense for structurally compatible factors.
standard math Theorem 3 of [14] is accepted as true: the epsilon-ACP error bound holds as stated.
The paper relies on this cited theorem to prove HACP's bounds (Prop. 4).
standard math Chan-Darwiche distance bounds (Eq. 4 and Eq. 7) are valid.
Used in Theorem 5 to bound maximal deviation.

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Cite this review

Pith. "Pith review of Compression versus Accuracy: A Hierarchy of Lifted Models." pith.science (2026). https://pith.science/paper/M5XMO3SK

@misc{pith2026250522288,
  author       = {Pith},
  title        = {Pith review of: Compression versus Accuracy: A Hierarchy of Lifted Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M5XMO3SK}},
  note         = {Machine review of arXiv:2505.22288}
}

abstract

Probabilistic graphical models that encode indistinguishable objects and relations among them use first-order logic constructs to compress a propositional factorised model for more efficient (lifted) inference. To obtain a lifted representation, the state-of-the-art algorithm Advanced Colour Passing (ACP) groups factors that represent matching distributions. In an approximate version using $\varepsilon$ as a hyperparameter, factors are grouped that differ by a factor of at most $(1\pm \varepsilon)$. However, finding a suitable $\varepsilon$ is not obvious and may need a lot of exploration, possibly requiring many ACP runs with different $\varepsilon$ values. Additionally, varying $\varepsilon$ can yield wildly different models, leading to decreased interpretability. Therefore, this paper presents a hierarchical approach to lifted model construction that is hyperparameter-free. It efficiently computes a hierarchy of $\varepsilon$ values that ensures a hierarchy of models, meaning that once factors are grouped together given some $\varepsilon$, these factors will be grouped together for larger $\varepsilon$ as well. The hierarchy of $\varepsilon$ values also leads to a hierarchy of error bounds. This allows for explicitly weighing compression versus accuracy when choosing specific $\varepsilon$ values to run ACP with and enables interpretability between the different models.

Figures

Figures reproduced from arXiv: 2505.22288 by the authors.

**Figure 1.** An exemplary FG encoding a full joint probability distribution over three randvars A, B, and C. Definition 2 (Query). A query P(Q | E1 = e1, . . . , Ek = ek) consists of a query term Q and a set of events {Ej = ej} k j=1 where Q and all Ej , j = 1, . . . , k, are randvars. To query a specific probability instead of a distribution, the query term is an event Q = q. Lifted inference exploits identical behaviour of i… view at source ↗

**Figure 2.** Exemplary visualisation of a factor ordering with increasing ε. This information is easily stored in the list L and is easily readable from the matrix Λ˜. The εi are ordered by size (ε1 being the smallest value of them). Note that a root ε is always the maximal ε-distance of all pairwise factor comparisons of all leafs. E.g., ε4 = max{ε1,2, ε1,3, ε1,4, ε2,3, ε2,4, ε3,4}. Factor ϕ1 ϕ2 · · · · · · ϕm−1 ϕm ϕ1 0 ε1,2 ε1… view at source ↗

**Figure 3.** Showing fupper and flower over [0, 1] with d2 values from Corollary 6 to use an upper estimate for pmax ∆ by bounding it from above. 0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 1.0 pmax∆ in dependence of a chosen ε ε pmax∆ m = 10 m = 100 m = 1000 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The bound on pmax ∆ depending on the choice of ε for different amounts of factors m. Corollary 6. With previous notations, the change in any probabilistic query in an initial model M and a modified model M′ obtained by running HACP (Alg. 2) or ε-ACP is bounded by pmax…

**Figure 5.** Figure 5: Bounds of Eq. (8) in comparison to p for different d values depending on m and ε (Eq. (5) of the main paper). Circles/ crosses are the maximum distances from those bounds to the function p. Distances to function f(p) = p are later also referred as f = fupper for p ∈ [0…

**Figure 6.** Figure 6: Showing the fupper and flower functions over [0, 1] with d2 values from Corollary 6 to use an upper estimate for pmax ∆ by bounding it from above. (see [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: The maximal choice of ε depending on the maximal deviation p ∗ ∆ for different amount of factors m to guarantee pmax ∆ ≤ p ∗ ∆ as proven in Theorem 7. Since q1 ≥ 0 ⇔ m−1 m ≥ p ∗ ∆, the minus option ε2 is smaller than 0 and knowing that m ≥ 2 already guarantees the resu…

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Works this paper leans on

24 extracted references · 24 canonical work pages

[14]

F. R. Kschischang, B. J. Frey, and H.-A. Loeliger. Factor Graphs and the Sum-Product Algorithm . IEEE Transactions on Information Theory , 47: 0 498--519, 2001

work page 2001
[1]

Chan and A

H. Chan and A. Darwiche. A distance measure for bounding probabilistic belief change. International Journal of Approximate Reasoning , 38:149--174, 2005

work page 2005
[3]

Ahmadi, K

B. Ahmadi, K. Kersting, M. Mladenov, and S. Natarajan. Exploiting Symmetries for Scaling Loopy Belief Propagation and Relational Training . Machine Learning, 92 0 (1): 0 91--132, 2013

work page 2013
[4]

Braun and R

T. Braun and R. M \"o ller. Lifted Junction Tree Algorithm . In Proceedings of the Thirty-Ninth German Conference on Artificial Intelligence (KI-2016), pages 30--42. Springer, 2016

work page 2016
[5]

Braun and R

T. Braun and R. M \"o ller. Parameterised Queries and Lifted Query Answering . In Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence (IJCAI-2018), pages 4980--4986. IJCAI Organization, 2018

work page 2018
[6]

Chan and A

H. Chan and A. Darwiche. A Distance Measure for Bounding Probabilistic Belief Change . International Journal of Approximate Reasoning, 38: 0 149--174, 2005

work page 2005
[7]

De Salvo Braz , E

R. De Salvo Braz , E. Amir, and D. Roth. Lifted First-Order Probabilistic Inference . In Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence (IJCAI-2005), pages 1319--1325. Morgan Kaufmann Publishers Inc., 2005

work page 2005
[8]

De Salvo Braz , E

R. De Salvo Braz , E. Amir, and D. Roth. MPE and Partial Inversion in Lifted Probabilistic Variable Elimination . In Proceedings of the Twenty-First National Conference on Artificial Intelligence (AAAI-2006), pages 1123--1130. AAAI Press, 2006

work page 2006

Show all 24 references

[9]

B. J. Frey, F. R. Kschischang, H.-A. Loeliger, and N. Wiberg. Factor Graphs and Algorithms . In Proceedings of the Thirty-Fifth Annual Allerton Conference on Communication, Control, and Computing, pages 666--680. Allerton House, 1997

1997
[10]

Gogate and P

V. Gogate and P. Domingos. Probabilistic Theorem Proving . In Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence (UAI-2011), pages 256--265. AUAI Press, 2011

2011
[11]

Hartwig, R

M. Hartwig, R. M \"o ller, and T. Braun. An Extended View on Lifting Gaussian Bayesian Networks . Artificial Intelligence, 330: 0 104082, 2024

2024
[12]

Kersting, B

K. Kersting, B. Ahmadi, and S. Natarajan. Counting Belief Propagation . In Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence (UAI-2009), pages 277--284. AUAI Press, 2009

2009
[13]

Kisy\' n ski and D

J. Kisy\' n ski and D. Poole. Constraint Processing in Lifted Probabilistic Inference . In Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence (UAI-2009), pages 293--302. AUAI Press, 2009

2009
[15]

Luttermann, T

M. Luttermann, T. Braun, R. Möller, and M. Gehrke. Colour Passing Revisited: Lifted Model Construction with Commutative Factors . In Proceedings of the Thirty-Eighth AAAI Conference on Artificial Intelligence (AAAI-2024), pages 20500--20507. AAAI Press, 2024

2024
[16]

Luttermann, J

M. Luttermann, J. Speller, M. Gehrke, T. Braun, R. Möller, and M. Hartwig. Approximate Lifted Model Construction . In Proceedings of the Thirty-Fourth International Joint Conference on Artificial Intelligence (IJCAI-2025), 2025. https://arxiv.org/abs/2504.20784

2025 arXiv
[17]

Milch, L

B. Milch, L. S. Zettlemoyer, K. Kersting, M. Haimes, and L. P. Kaelbling. Lifted Probabilistic Inference with Counting Formulas . In Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence (AAAI-2008), pages 1062--1068. AAAI Press, 2008

2008
[18]

Niepert and G

M. Niepert and G. Van den Broeck. Tractability through Exchangeability: A New Perspective on Efficient Probabilistic Inference . In Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence (AAAI-2014), pages 2467--2475. AAAI Press, 2014

2014
[19]

D. Poole. First-order Probabilistic Inference . In Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence (IJCAI-2003), pages 985--991. IJCAI Organization, 2003

2003
[20]

Richardson and P

M. Richardson and P. Domingos. Markov Logic Networks . Machine Learning, 62 0 (1--2): 0 107--136, 2006

2006
[21]

Taghipour, D

N. Taghipour, D. Fierens, J. Davis, and H. Blockeel. Lifted Variable Elimination: Decoupling the Operators from the Constraint Language . Journal of Artificial Intelligence Research, 47 0 (1): 0 393--439, 2013

2013
[22]

Van den Broeck

G. Van den Broeck. Lifted Inference and Learning in Statistical Relational Models . PhD thesis, KU Leuven, 2013

2013
[23]

Van den Broeck and J

G. Van den Broeck and J. Davis. Conditioning in First-Order Knowledge Compilation and Lifted Probabilistic Inference . In Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence (AAAI-2012), pages 1961--1967. AAAI Press, 2012

2012
[24]

Van den Broeck, N

G. Van den Broeck, N. Taghipour, W. Meert, J. Davis, and L. De Raedt. Lifted Probabilistic Inference by First-order Knowledge Compilation . In Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence (IJCAI-2011), pages 2178--2185. IJCAI Organ...

2011
[25]

Weisfeiler and A

B. Weisfeiler and A. A. Leman. The Reduction of a Graph to Canonical Form and the Algebra which Appears Therein . NTI , Series , 2: 0 12--16, 1968. English translation by Grigory Ryabov available at https://www.iti.zcu.cz/wl2018/pdf/wl_paper_translation.pdf

1968

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [14]

F. R. Kschischang, B. J. Frey, and H.-A. Loeliger. Factor Graphs and the Sum-Product Algorithm . IEEE Transactions on Information Theory , 47: 0 498--519, 2001

work page 2001

[2] [1]

Chan and A

H. Chan and A. Darwiche. A distance measure for bounding probabilistic belief change. International Journal of Approximate Reasoning , 38:149--174, 2005

work page 2005

[3] [3]

Ahmadi, K

B. Ahmadi, K. Kersting, M. Mladenov, and S. Natarajan. Exploiting Symmetries for Scaling Loopy Belief Propagation and Relational Training . Machine Learning, 92 0 (1): 0 91--132, 2013

work page 2013

[4] [4]

Braun and R

T. Braun and R. M \"o ller. Lifted Junction Tree Algorithm . In Proceedings of the Thirty-Ninth German Conference on Artificial Intelligence (KI-2016), pages 30--42. Springer, 2016

work page 2016

[5] [5]

Braun and R

T. Braun and R. M \"o ller. Parameterised Queries and Lifted Query Answering . In Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence (IJCAI-2018), pages 4980--4986. IJCAI Organization, 2018

work page 2018

[6] [6]

Chan and A

H. Chan and A. Darwiche. A Distance Measure for Bounding Probabilistic Belief Change . International Journal of Approximate Reasoning, 38: 0 149--174, 2005

work page 2005

[7] [7]

De Salvo Braz , E

R. De Salvo Braz , E. Amir, and D. Roth. Lifted First-Order Probabilistic Inference . In Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence (IJCAI-2005), pages 1319--1325. Morgan Kaufmann Publishers Inc., 2005

work page 2005

[8] [8]

De Salvo Braz , E

R. De Salvo Braz , E. Amir, and D. Roth. MPE and Partial Inversion in Lifted Probabilistic Variable Elimination . In Proceedings of the Twenty-First National Conference on Artificial Intelligence (AAAI-2006), pages 1123--1130. AAAI Press, 2006

work page 2006