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REVIEW 5 major objections 5 minor 35 references

Hybrid Causal Identification and Causal Mechanism Clustering

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A mixture of additive noise models is identifiable from pooled observational data, so comparing variational likelihood bounds in the two directions finds the true causal direction, and clustering the residual reveals the generating…

desk verdict The algorithmic combination is interesting and the empirical results are strong, but the identifiability proof has a fatal algebraic gap, so the causal decision rule currently rests on sand. read the letter →

arxiv 2507.21792 v1 pith:EPDE24HK submitted 2025-07-29 cs.AI

Saixiong Liu , Yuhua Qian , Jue Li , Honghong Cheng , Feijiang Li This is my paper
classification cs.AI
keywords hybridadditivenoisemodelbivariatecausaldiscoverymixtureconditionalvariationalautoencoderELBOlikelihoodcriterionmechanismclusteringheterogeneousobservationaldataidentifiabilityinference
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

The paper claims that bivariate causal direction remains identifiable when the data come from a mixture of additive noise mechanisms, not just a single mechanism. It constructs a mixture conditional variational auto-encoder whose evidence lower bound approximates the conditional likelihood in each direction, and decides X→Y versus Y→X by whichever direction has the larger bound. In the chosen direction, the residual noise term carries the identity of the generating mechanism, so clustering that residual separates samples by cause rather than by geometric proximity. If these claims hold, heterogeneous observational data pooled across environments or locations—without environment labels—can be used for causal discovery and for clustering data by causal mechanism.

What carries the argument

The load-bearing object is the Hybrid Additive Noise Model (HANM), a weighted sum of $K$ additive noise mechanisms, together with the mixture conditional variational auto-encoder built to fit it. The identifiability argument reduces the existence of a reverse HANM to a third-order ordinary differential equation (Eq. 7) on $\xi = \log p(X)$: generically no reverse model can satisfy it, so the likelihood gap between the ELBO bounds (Eqs. 25 and 26) is the decision rule. For clustering, the residual $w\epsilon$ of the chosen direction is the extracted causal feature space, and the clustering objective minimizes the squared distance from residuals to cluster centers in that space.

What would settle it

Generate data from a known forward HANM with two near-linear components and small noise, then run MCVCI many times; if the forward and reverse ELBO scores are statistically indistinguishable across seeds, the reverse-identifiability premise fails.

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Extended reading notes

Core claim

The central discovery is a generic identifiability result for the Hybrid Additive Noise Model (HANM), written as $Y = \sum_{k=1}^K w_k(f_k(x_k) + \epsilon_k)$ with $\sum w_k = 1$ and $x_k \perp\!\!\perp \epsilon_k$. The paper proves that if a forward HANM holds, a reverse HANM $X = \sum_{k=1}^K w_k(g_k(y_k) + \tilde\epsilon_k)$ can exist only when the log-density of $X$ satisfies a specific third-order ordinary differential equation constraint; generically no such reverse model exists. Building on this, the mixture conditional variational auto-encoder maximizes an ELBO that lower-bounds $\log p(Y|X)$, and the same bound in the reverse direction lower-bounds $\log p(X|Y)$; comparing the two scores gives the causal direction. The residual $w\epsilon$ in the winning direction is then used as the feature space for causal mechanism clustering.

Load-bearing premise

The proof that a reverse hybrid additive noise model is almost impossible collapses the mixture's $K$ components into single functions in Eqs. (10)-(11) and asserts genericity without a formal measure argument, so if the flexible neural regressor can approximate the reverse direction anyway, the two likelihood scores become nearly equal and the direction decision loses its basis.

Editorial extensions

If this is right

  • Bivariate causality on pooled heterogeneous data can be decided without knowing which environment each sample came from.
  • Cluster assignments can be derived from the data-generating mechanism rather than from raw feature distance, separating shifts in function or noise.
  • The likelihood gap defines a confidence score $\tau$, so users can set a threshold and abstain on low-confidence pairs instead of committing to a direction.
  • The variational regressor does not require Gaussian noise within a component, so the method covers nonlinear functions and non-Gaussian additive noise.
  • On the SIM, SIM-G, SIM-ln, and cause-effect-pair benchmarks the reported direction accuracy exceeds the comparison methods, and on the simulated and air-quality clustering tasks the reported ARI and NMI exceed the baselines.

Reading between the lines

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

  • A direct stress test the paper does not run: generate near-linear, low-noise two-component mixtures where the reverse regression is almost as simple as the forward one, and check whether the two ELBO scores become statistically indistinguishable; if they do, the generic identifiability claim needs a measure-theoretic qualification.
  • The residual-clustering idea suggests a practical extension: the same trained model could detect unknown environment switches or label shifts by inspecting whether residuals naturally split into more clusters than the assumed $K$.
  • Using the confidence threshold $\tau$ as an abstention rule in downstream applications would convert the top-k decision-rate curve into a deployable policy, but the paper does not propose such a procedure.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper proposes MCVCI, a method for bivariate causal direction identification on heterogeneous observational data, and MCVCC, an extension for clustering causal mechanisms. The model is a mixture of additive noise models (HANM); the authors claim an identifiability result (Lemma 1/Theorem 1) stating that a reverse HANM is 'almost impossible', and use it to justify a likelihood-ratio decision rule computed via a mixture conditional variational auto-encoder (CVAE). MCVCC clusters the residuals of the selected direction. Experiments on SIM/SIM-G/SIM-ln, CEP, and BAFU air data report accuracy/ARI/NMI improvements over several baselines, including ANM-MM, CANM, and classical methods. The central theoretical claim, however, rests on a proof that contains elementary algebraic errors in the log-density and derivative calculations, and the 'almost impossible' assertion lacks any measure-theoretic argument. The decision rule therefore lacks a valid identifiability foundation, and the empirical results, while reported as favorable, are not connected to a sound theoretical justification.

Significance. If the HANM identifiability theorem and the ELBO-based decision rule were correct, the paper would offer a practically useful extension of additive-noise causal inference to heterogeneous data, and the clustering application would be a modest novel contribution. The experimental coverage is broad, with several external benchmarks (SIM, SIM-G, SIM-ln, CEP) and a comparison against many recent methods, including ANM-MM and CANM. These strengths are real and deserve credit. However, the theoretical pillar—the identifiability of the HANM—is not established: the proof of Lemma 1 replaces the logarithm of a mixture by a weighted sum of logarithms, collapses sums of component derivatives into single functions without justification, and asserts genericity without a measure. Since the likelihood-ratio criterion depends entirely on this theorem, the central claim is unsupported. The paper does not provide machine-checked proofs or reproducible code, and the reported performance cannot compensate for the absence of a valid theoretical guarantee.

major comments (5)
  1. [Section III-C, Eq. (8)] The proof begins by writing the log-joint in the reverse model as π(X,Y)=Σ_k w_k v_k(y_k−f_k(x_k)) + ξ(X). But p(X,Y) is a mixture density, so log p(X,Y)=ξ(X)+log Σ_k w_k p_{ε_k}(y_k−f_k(x_k)). The paper replaces the log of the sum with the weighted sum of logs, which is not an identity. Every subsequent derivative in Eqs. (9)–(11) is therefore taken from an expression that is not the actual log-density; the derived ODE (7) and the conclusion of Lemma 1 do not follow.
  2. [Section III-C, Eq. (10)] Equation (10) collapses Σ_k w_k ev''_k(x_k−g_k(y_k)) g'_k(y_k) into a single function ev''(X−g(Y)). This step is valid only if all component functions g_k are identical (or under other unstated conditions that force the sum to form a single function of (X,Y)). No such condition is given. The same issue affects Eq. (11), where Σ_k w_k ev''_k is replaced by a single ev''. The derivation of Eq. (12) and the subsequent cancellation steps depends on this unjustified collapse, so the proof of Lemma 1 is incomplete at a load-bearing point.
  3. [Section III-C, Theorem 1] The theorem concludes that 'it is almost impossible to exist a hybrid ANM satisfying the condition from Y→X' and states that the set of solutions log p_X is contained in a three-dimensional affine space. No measure-theoretic argument is provided: there is no definition of the space of distributions or functions over which a measure is being taken, no statement of the genericity assumption, and no proof that the constraint set has measure zero. The 'almost impossible' claim is asserted rather than demonstrated. Since the likelihood-ratio decision rule in Eqs. (25)–(26) is justified solely by this genericity, the rule lacks a theoretical guarantee.
  4. [Section III-B and III-D, Eqs. (3), (5), (22)] The generative model is internally inconsistent. Equation (3) defines Y as a weighted sum of component functions, Y=Σ_k w_k(f_k(x_k)+ε_k), which is a single functional relationship. In contrast, Eq. (5) and the subsequent proof treat p(Y|X) as a mixture density Σ_k w_k p_{ε_k}(...), and Eq. (22) again claims log p(Y|X)=Σ_k w_k log p_{ε_k}(...). A weighted sum of functions is not the same as a mixture of conditional densities, and the logarithm of a mixture is not the mixture of logarithms. This inconsistency affects both the identifiability proof and the ELBO derivation: the variational model in Fig. 1 and Eq. (23) uses a mixture decomposition, while the causal criterion in Eqs. (25)–(26) is written as a sum of log-likelihoods. The relationship between the fitted CVAE and the stated HANM is therefore unclear, and the clustering objective in Eq. (27), which uses the residual from the selected direction, inherits this ambiguity.
  5. [Section III-D, Eqs. (23)–(26)] The ELBO in Eq. (24) is a lower bound for log pθ(Y|X) under a specific conditional generative model, but the paper does not show that this lower bound is a faithful proxy for the HANM marginal log-likelihood when comparing the two directions. In particular, the KL term and the encoder distribution qφ are optimized separately for the forward and reverse models, and the comparison in Algorithm 1 (step 7) uses the sum log p(X)+ELBO vs. log p(Y)+ELBO. Without an analysis of the approximation gap, two nearly equal ELBOs can arise even when the true likelihoods differ; conversely, a flexible VAE might approximate the reverse direction well, making the scores close and the decision unreliable. The empirical confidence analysis in Section IV-A4 measures only the decision rate of the top k%, not the validity of the underlying likelihood comparison.
minor comments (5)
  1. [Section IV-A4] The confidence paragraph contains a duplicated sentence: 'The higher the value of τ, the more correct our decision will be. Furthermore, we can set a threshold t to require τ ≥ t.' appears twice in a row. This should be cleaned up.
  2. [Throughout] Several typos and nonstandard notations occur, e.g., 'casual' for 'causal' in the abstract and Algorithm 1 outputs, 'eϵ' vs. 'ε' inconsistency, and 'wϵc' in Section III-E. Please proofread carefully.
  3. [Table IV] The header of Table IV repeats 'DIFFERENT CLUSTER NUMBERS CONDITION' from Table III; the actual setting (BAFU air data) should be stated in the caption.
  4. [References] The reference list is incomplete or inconsistent in places: reference [32] is formatted as 'S., Hoyer, et al.' with an unusual author order, and several cited works (e.g., the ANM-MM paper, the PNL paper) would benefit from full bibliographic data.
  5. [Section IV-B2] The description of the constructed simulated datasets is terse: it does not specify how the component index k is assigned to samples, how many samples per component are drawn, or how the mixture weights w_k are chosen. Please provide the full data-generation protocol so that the clustering experiments can be reproduced.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the derivation chain is self-contained and the empirical evaluation is against external benchmarks.

full rationale

The paper's central derivation is an identifiability theorem for the Hybrid Additive Noise Model, followed by an ELBO-based likelihood comparison and a clustering method on the residual of the chosen direction. No step reduces to its own inputs by construction. The identifiability argument (Lemma 1 and Theorem 1, Eqs. 7-21) is derived internally from differential equations and genericity reasoning, not from a fitted parameter or from the paper's own prior results. The likelihood decision rule (Eqs. 25-26) is applied to held-out test sets on public benchmarks (SIM, SIM-G, SIM-ln, CEP), and the clustering method (MCVCC) is evaluated against ground-truth labels on synthetic data and the BAFU dataset; cluster labels are not used during training, so the empirical claims are self-contained. The residual features used for clustering are model outputs, not fitted parameters, so there is no fitted-input-called-prediction circularity. While the proof of Theorem 1 contains algebraic steps that appear mathematically incorrect (e.g., Eq. (8) replaces the log of a mixture density with a weighted sum of component log-densities), that is a correctness risk, not circularity, and is outside the scope of this analysis. No self-citations are load-bearing. Score 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on several unproved modeling assumptions: the data follow a HANM with known K; the reverse HANM is generically impossible; the ELBO is a valid proxy for the true conditional likelihood; and the residual of the chosen direction separates mechanisms. No new physical entities are introduced. The free parameters K and C are chosen by the user or by model selection, and the threshold t is a free decision parameter.

free parameters (3)
  • K (number of mixture components) = Not reported; selected per dataset during training
    MCVCI and MCVCC both require K, the number of HANM components; the paper says an appropriate K is selected through training but does not give values or a validation procedure.
  • C (number of clusters in MCVCC) = 2 (f1-f5, BAFU), 3 and 4 in Table III experiments
    MCVCC's clustering objective (Eq. 27) requires C; in all experiments C equals the true number of classes, so the method is not tested with unknown C.
  • Confidence threshold t = Not specified
    The confidence rule tau >= t introduces a decision threshold that can be tuned; the paper discusses it but does not fix or evaluate it.
assumptions (4)
  • domain assumption Observed data are generated by a HANM with a finite, known number of components K, with each component an additive noise model.
    Section III-B defines the model; the likelihood criterion and clustering features are only meaningful under this generative assumption.
  • ad hoc to paper A reverse HANM is 'almost impossible': the set of joint distributions admitting both forward and reverse HANM has measure zero.
    Lemma 1/Theorem 1 assert this genericity, but the proof is incomplete (Eqs. 10-11 merge component sums unjustifiably) and no measure argument is given. The causal decision rule relies on this genericity.
  • ad hoc to paper The ELBO of the mixture conditional VAE is a faithful proxy for the true conditional log-likelihood when comparing X->Y vs Y->X.
    Section III-D defines LX->Y and LY->X as marginal log-density plus ELBO; the paper does not prove the lower bounds are tight enough to preserve the true causal ordering in both directions.
  • domain assumption The residual theta = X - eX or Y - eY from the selected causal direction is a sufficient feature for causal mechanism clustering.
    Algorithm 2 clusters on theta with k-means; the paper only shows empirical success on constructed data, not a derivation that theta captures the mechanism identity.

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Pith. "Pith review of Hybrid Causal Identification and Causal Mechanism Clustering." pith.science (2026). https://pith.science/paper/EPDE24HK

@misc{pith2026250721792,
  author       = {Pith},
  title        = {Pith review of: Hybrid Causal Identification and Causal Mechanism Clustering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EPDE24HK}},
  note         = {Machine review of arXiv:2507.21792}
}
read the original abstract

Bivariate causal direction identification is a fundamental and vital problem in the causal inference field. Among binary causal methods, most methods based on additive noise only use one single causal mechanism to construct a causal model. In the real world, observations are always collected in different environments with heterogeneous causal relationships. Therefore, on observation data, this paper proposes a Mixture Conditional Variational Causal Inference model (MCVCI) to infer heterogeneous causality. Specifically, according to the identifiability of the Hybrid Additive Noise Model (HANM), MCVCI combines the superior fitting capabilities of the Gaussian mixture model and the neural network and elegantly uses the likelihoods obtained from the probabilistic bounds of the mixture conditional variational auto-encoder as causal decision criteria. Moreover, we model the casual heterogeneity into cluster numbers and propose the Mixture Conditional Variational Causal Clustering (MCVCC) method, which can reveal causal mechanism expression. Compared with state-of-the-art methods, the comprehensive best performance demonstrates the effectiveness of the methods proposed in this paper on several simulated and real data.

Figures

Figures reproduced from arXiv: 2507.21792 by the authors.

Figure 1
Figure 1. Regression model of the mixture conditional variational auto-encoder. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Decision rate curves of MCVCI under the top [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. The clustering results of MCVCC and comparison algorithms when [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: The clustering results of MCVCC and comparison algorithms when [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: The clustering results of MCVCC and comparison algorithms when [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: The clustering results of MCVCC and comparison algorithms when [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: The clustering results of MCVCC and comparison algorithms when [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: The clustering results of MCVCC and comparison algorithms when [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: The clustering results of MCVCC and comparison algorithms when [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: The clustering results of MCVCC and comparison algorithms when [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: Clustering results of comparison algorithms and MCVCC on BAFU air data. [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

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Works this paper leans on

35 extracted references · 32 canonical work pages

  1. [1]

    Domain adaptation under structural causal models,

    Chen, Yuansi, and Peter B ¨uhlmann, “Domain adaptation under structural causal models,” J. Mach. Learn. Res. , vol. 22, no. 1, pp. 11856-11935, 2021. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 8 TABLE IV CLUSTERING RESULTS OF COMPARISON METHODS AND MCVCC ON DIFFERENT CLUSTER NUMBERS CONDITION. k-means SpeClu GMM CV AE-km ANM-MM MCVCC ARI -0...

    work page 2021

  2. [2]

    Stable learning establishes some common ground between causal inference and ma-chine learning,

    Cui, Peng, and Susan Athey, “Stable learning establishes some common ground between causal inference and ma-chine learning,” Nat. Mach. Intell., vol. 4, no. 2, pp. 110-115, 2022

    work page 2022

  3. [3]

    Causal discovery on discrete data with extensions to mixture model,

    Liu, Furui, and Laiwan Chan, “Causal discovery on discrete data with extensions to mixture model,” ACM Trans. Intell. Syst. Technol. , vol. 7, no. 2, pp. 1-19, 2015

    work page 2015

  4. [4]

    Pearl, Judea, Causality, Cambridge university press, 2009

    work page 2009

  5. [5]

    On the identifiability of the post- nonlinear causal model,

    Zhang, Kun, and Aapo Hyvarinen, “On the identifiability of the post- nonlinear causal model,” arXiv:1205.2599, 2012

    arXiv 2012

  6. [6]

    Nonlinear causal discovery with additive noise models,

    Hoyer, Patrik, et al., “Nonlinear causal discovery with additive noise models,” Adv. Neural Inf. Process. Syst. , vol. 21, 2008

    work page 2008

  7. [7]

    Information-geometric ap- proach to inferring causal directions,

    Janzing, Dominik, and Bernhard Sch ¨olkopf, “Information-geometric ap- proach to inferring causal directions,” J. Artif. Intell. , vol. 182, pp. 1- 31,2012

    work page 2012

  8. [8]

    Cause-effect inference by comparing regression errors,

    Bl ¨obaum, Patrick, et al, “Cause-effect inference by comparing regression errors,” in Proc. Int. Artif. Intell. Statist. , pp. 900-909, 2018

    work page 2018

Show all 35 references
  1. [9]

    Adversarial orthogonal regression: Two non-linear regressions for causal inference,

    Heydari, M. Reza, Saber Salehkaleybar, and Kun Zhang, “Adversarial orthogonal regression: Two non-linear regressions for causal inference,” Neural Netw., vol. 143, pp. 66-73, 2021

    2021

  2. [10]

    Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery,

    Tagasovska, Natasa, Val ´erie Chavez-Demoulin, and Thibault Vatter, “Distinguishing cause from effect using quantiles: Bivariate quantile causal discovery,” in Proc. Int. Conf. Mach Learn. , PMLR, 2020

    2020

  3. [11]

    Causal Inference with Heteroscedastic Noise Mod- els,

    Xu, Sascha, et al., “Causal Inference with Heteroscedastic Noise Mod- els,” in Proc. of AAAI Workshop Inf. Theoretic Causal Inference and Discovery, 2022

    2022

  4. [12]

    Causal inference and mechanism clustering of a mixture of additive noise models,

    Hu, Shoubo, et al., “Causal inference and mechanism clustering of a mixture of additive noise models,” Adv. Neural Inf. Process. Syst. , vol. 31, 2018

    2018

  5. [13]

    Pearl, Judea, Models, reasoning and inference , Cambridge, UK: Cam- bridge University Press, vol. 19, no. 2, pp. 3, 2000

    2000

  6. [14]

    Multi-source causal feature selection,

    Yu, Kui, et al., “Multi-source causal feature selection,” IEEE Trans. Pattern Anal. Mach. Intell. , vol. 42, no. 9, pp. 2240-2256, 2019

    2019

  7. [15]

    Learning structured output representation using deep conditional generative models,

    Sohn, Kihyuk, Honglak Lee, and Xinchen Yan, “Learning structured output representation using deep conditional generative models,” Adv. Neural Inf. Process. Syst. , vol. 28, 2015. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 9 (a) (b) (c) (d) (e) (f) (g) Fig. 5. Th...

    2015

  8. [16]

    Identifying cause and effect on discrete data using additive noise models,

    Peters, Jonas, Dominik Janzing, and Bernhard Sch ¨olkopf. “Identifying cause and effect on discrete data using additive noise models,” in Proc. 9th Int. Workshop Artif. Intell. Statist. , 2010

    2010

  9. [17]

    Telling cause from effect using MDL-based local and global regression,

    Marx, Alexander, and Jilles Vreeken. “Telling cause from effect using MDL-based local and global regression,” in Proc. IEEE Int.Conf. Data Mining, IEEE, pp. 307-316, 2017

    2017

  10. [18]

    Identifiability of cause and effect using regularized regression,

    Marx, Alexander, and Jilles Vreeken. “Identifiability of cause and effect using regularized regression,” in Proc. 25th ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining , pp. 852-861, 2019

    2019

  11. [19]

    Causal network learning with non- invertible functional relationship,

    Wang, Bingling, and Qing Zhou, “Causal network learning with non- invertible functional relationship,” Comput. Stat. Data Anal., vol. 156, pp. 107141, 2021

    2021

  12. [20]

    Causal discov- ery toolbox: Uncovering causal relationships in python,

    Kalainathan, Diviyan, Olivier Goudet, and Ritik Dutta, “Causal discov- ery toolbox: Uncovering causal relationships in python,” J. Mach. Learn. Res., vol. 21, no. 1, pp.1406-1410, 2020

    2020

  13. [21]

    Distance-based clustering of CGH data,

    Liu, Jun, et al., “Distance-based clustering of CGH data,” Bioinformat- ics, vol. 22, no. 16, pp. 1971-1978, 2006

    1971

  14. [22]

    Causal discovery with continuous additive noise models,

    Peters, Jonas, et al., “Causal discovery with continuous additive noise models,” J. Mach. Learn. Res. , 2014

    2014

  15. [23]

    Causal discovery with cascade nonlinear additive noise models,

    Cai, Ruichu, et al., “Causal discovery with cascade nonlinear additive noise models,” arXiv:1905.09442, 2019

    1905 arXiv

  16. [24]

    Distinguishing cause from effect using obser- vational data: methods and benchmarks,

    Mooij, Joris M., et al., “Distinguishing cause from effect using obser- vational data: methods and benchmarks,” J. Mach. Learn. Res. , vol. 17, no. 1, pp. 1103-1204, 2016

    2016

  17. [25]

    Inference of cause and effect with unsupervised inverse regression,

    Sgouritsa, Eleni, et al., “Inference of cause and effect with unsupervised inverse regression,” Artif. Intell. Statist. , PMLR, 2015

    2015

  18. [26]

    Jonas Peters

    Peter B ¨uhlmann. Jonas Peters. Jan Ernest. ”CAM: Causal additive models, high-dimensional order search and penalized regression.” Ann. Statist., 42 (6) 2526 - 2556, December 2014. https://doi.org/10.1214/14- AOS1260

    2014 doi

  19. [27]

    Some methods for classification and analysis of multivariate observations,

    MacQueen, James, “Some methods for classification and analysis of multivariate observations,” in Proc. Fifth Berkeley Symp. Math. Statist. and Probab., vol. 1, no. 14, 1967

    1967

  20. [28]

    On spectral clustering: Analysis and an algorithm,

    Ng, Andrew, Michael Jordan, and Yair Weiss., “On spectral clustering: Analysis and an algorithm,” in Adv. Neural Inf. Process. Syst. , vol. 14, 2001

    2001

  21. [29]

    The infinite Gaussian mixture model,

    Rasmussen, Carl, “The infinite Gaussian mixture model,” in Adv. Neural Inf. Process. Syst. , vol. 12, 1999

    1999

  22. [30]

    Normalized cuts and image segmenta- tion,

    Shi, Jianbo, and Jitendra Malik, “Normalized cuts and image segmenta- tion,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 22, no. 8, pp. 888-905, 2000

    2000

  23. [31]

    Comparing partitions,

    Hubert, Lawrence, and Phipps Arabie, “Comparing partitions,” J. Clas- sif., vol. 2, pp. 193-218, 1985

    1985

  24. [32]

    A linear non-Gaussian acyclic model for causal discovery,

    S., Hoyer, et al.,“A linear non-Gaussian acyclic model for causal discovery,” J. Mach. Learn. Res. , vol. 7, no. 10, 2006

    2006

  25. [33]

    Auto-encoding variational bayes,

    Kingma, Diederik P., and Max Welling, “Auto-encoding variational bayes,” arXiv:1312.6114, 2013

    2013 arXiv

  26. [34]

    Explanation in artificial intelligence: Insights from the social sciences,

    Miller, Tim, “Explanation in artificial intelligence: Insights from the social sciences,” Artif. Intell., vol. 267, pp. 1-38, 2019. Saixiong Liu Saixiong Liu is currently pursuing the Ph.D.degree at the School of Computer Science and Information Engineering and working at the ...

    2019

  27. [2020]

    She was a visiting scholar with the City University of Hong Kong, Hong Kong, China, in 2019

    She is currently a teacher with the School of Information, Shanxi University of Finance and Economics. She was a visiting scholar with the City University of Hong Kong, Hong Kong, China, in 2019. Her research interests include associations mining, multimodal learning, data min...

    2019

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