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This paper proves that partial-mastery cognitive diagnostic models are locally identifiable under Q-matrix conditions requiring at least three pure items per attribute, so any two parameter arrays producing the same response distribution di

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2026-08-01 20:33 UTC pith:SQOCFE3Y

load-bearing objection First identifiability theorems for PM-CDMs; the proof has a repairable but real gap in the transfer to the copula-restricted parameterization, plus a model-specification inconsistency around marginal uniformness. the 3 major comments →

arxiv 2607.16593 v1 pith:SQOCFE3Y submitted 2026-07-18 math.ST stat.TH

Identifiability of Partial-Mastery Cognitive Diagnostic Models

Jun Wu , Patr\'icia Martinkov\'a , Elena Erosheva This is my paper
classification math.ST stat.TH
keywords partial-mastery cognitive diagnostic modelsidentifiabilityQ-matrixGaussian copulaJacobian rankrestricted latent class modelsreal analytic functionsgeneric finite-to-one
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Partial-mastery cognitive diagnostic models (PM-CDMs) replace the usual binary mastery/non-mastery attribute indicators with continuous mastery scores, which makes them more flexible but leaves open whether their parameters can be inferred from response data. The paper establishes the first identifiability results for these models: if each latent attribute is measured by at least three 'pure' items that depend on that attribute alone — or, in a relaxed version, one attribute gets a third pure item and the rest get two — then the item parameters and the Gaussian-copula parameters of the mastery-score distribution are locally identifiable. That means the parameterization map is generically finite-to-one: no continuous family of different parameter values can explain the same data, and any ambiguity is confined to isolated alternatives. The proof reduces the model's enormous latent space to a low-dimensional algebraic form, verifies the Jacobian rank on a 5-item/2-attribute submodel by symbolic computation, and then exports that conclusion to the full model using real analyticity of Gaussian copula integrals. A sympathetic reader should care because without identifiability, estimation and inference in PM-CDMs would be ill-posed, and the paper also gives applied test-design guidance.

Core claim

The central claim is that, under the stated Q-matrix conditions, the PM-CDM parameterization map is generically finite-to-one: for almost every parameter value, no continuous curve of alternative item parameters and copula parameters produces the same response distribution; only finitely many isolated alternative points can exist. The paper establishes this by proving, in the minimal case of two attributes and five items with three pure items for one attribute and two for the other, that the Jacobian of a reduced parameterization has full rank at a point, and then shows the relevant minor is a nonzero real-analytic function on the Gaussian copula parameter space, so it vanishes only on a mea

What carries the argument

The central machinery is the 'reduced latent class probability' representation: latent response patterns are grouped by how many realized ones they contain in each block of items that require the same attributes, collapsing 2^{JK} classes into a low-dimensional parameter vector. This makes the model a rational map whose Jacobian can be studied. Three tools carry the proof: the T-matrix decomposition relating item-level response probabilities to joint response patterns; real analyticity of Gaussian copula integrals, which lets a nonzero Jacobian minor at a single point imply full rank almost everywhere on the copula submanifold; and a Gram-matrix argument that the functions generating later r

Load-bearing premise

The argument banks on the premise that one parameter point where the Jacobian has full rank is enough to conclude that the Gaussian-copula-restricted parameterization is generically finite-to-one, even though full rank at a single point does not by itself preclude a positive-dimensional fiber elsewhere on the copula submanifold.

What would settle it

For the minimal 5-item/2-attribute design, numerically search the parameter space for a second, distinct parameter point (or a curve) that yields the same vector of 32 response-pattern probabilities as a generic point; finding one would refute the claimed generic finite-to-one identifiability. Alternatively, compute the generic fiber dimension of the reduced map via Gröbner bases to see whether it is zero.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Estimation and inference for PM-CDMs are justified whenever the Q-matrix contains the required pure-item structure.
  • Assessment designers now have a concrete sufficiency criterion: at least three pure items per attribute, or two per attribute plus one extra pure item for one attribute.
  • The identifiability result extends to additive PM-CDMs, so that recently proposed subclass also carries the guarantee.
  • Unlike binary CDMs, PM-CDMs are shown to be at best generically finite-to-one, not globally one-to-one, under analogous conditions.
  • Failure to meet the conditions does not imply non-identifiability; the conditions are sufficient, not necessary.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The proof technique suggests the pure-item count conditions are likely not sharp; analogous Jacobian computations for other small Q-matrices could reveal identifiability with fewer pure items.
  • The real-analytic transfer argument should extend to other latent-distribution families with analytic density transforms, so the qualitative conclusion may hold beyond Gaussian copulas.
  • If global identifiability is sought next, the minimal 5-item/2-attribute example is the natural testbed: characterizing all preimages of its map would show whether finite ambiguity is actually small or grows with item count.
  • The finite-to-one result hints at an underlying symmetry or label-switching structure; characterizing that structure could lead to practical rules for choosing among equivalent parameter values.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper establishes the first identifiability results for partial-mastery cognitive diagnostic models (PM-CDMs). The main theorem states that the item parameters {θ_j,α} and Gaussian copula parameters (μ,Σ) are locally identifiable (in the algebraic-statistics sense of generic finite-to-one) when (i) K=1 and J≥4, or (ii) K≥2 and the Q-matrix contains three copies of I_K, i.e., at least three pure items per attribute. A second theorem relaxes the condition to one attribute having a third pure item. The proof uses a minimal K=2, J=5 submodel, a Macaulay2 Jacobian computation at a single point, real analytic continuation, and Xu's T-matrix techniques to propagate identifiability to the full model. The paper also extends the result to additive PM-CDMs and applies it to the ECPE Q-matrix.

Significance. If correct, this is a meaningful first step in the identifiability theory of PM-CDMs, directly extending classical CDM results (e.g., Xu 2017) to continuous latent mastery variables. The paper is commendably explicit about the non-algebraic nature of the copula parameterization and includes reproducible Macaulay2 code. The structural conditions are transparent and practically interpretable. However, the central proof contains a substantial gap in the transfer from a full-rank Jacobian of an unconstrained reparameterization to finite-to-one identifiability of the actual PM-CDM parameterization. Because this transfer is load-bearing, the main theorem is not established as written.

major comments (3)
  1. [§4.2.2 (Real Analytic Functions)] The inference from a nonzero 21×21 minor of the unconstrained Jacobian J(φ) (31×21 in (Θ,p^u)) to local identifiability of the PM-CDM submodel is invalid. The actual PM-CDM submodel has 15 parameters (10 θ, 5 copula), not 21. Its Jacobian is J(φ)·D, where D = ∂p^u/∂(μ,Σ) is an 11×5 matrix. The paper never computes D or its rank, nor shows that the p^u-directions are transverse to the θ-columns. Nonvanishing of a 21×21 minor of J(φ) does not imply that the 31×15 composed Jacobian has rank 15. The sentence 'Consequently, the Jacobian matrix has full column rank almost everywhere on this subspace' refers to the wrong matrix. This is a load-bearing gap in Step 1.
  2. [§3.1/§4.2.2] Even if a full-rank Jacobian were established almost everywhere on Ω′, the paper's conclusion 'local identifiability' (generic finite-to-one) would not follow automatically. Sullivant's Proposition 16.1.7 applies to rational maps; the PM-CDM parameterization involves Gaussian CDFs and is not rational. Full column rank at a point gives local injectivity in a neighborhood, not finiteness of global fibers. The paper does not prove that the exceptional set where fibers have positive dimension is contained in the rank-deficient set for the composed map. The real-analyticity argument only shows V∩Ω′ has measure zero for V = {rank J(φ)<21}, which is not sufficient.
  3. [§4.4.1 (PM-DINA)] The determinant condition for solving (4.3)–(4.4) reduces to (p00 p11 − p01 p10)(θ11−θ10) ≠ 0. The paper asserts that when item j shares a required latent attribute with item 1, 'the corresponding realized binary indicators o1 and oj are generally dependent' and hence p00 p11 − p01 p10 ≠ 0. This is not proven: for particular values of (μ,Σ), the Gaussian copula can yield independence even when both indicators depend on a common attribute. Since the identification of θ_j0 and θ_j1 in Step 3 relies on this determinant being nonzero, the argument needs a proof that the zero set has measure zero in the parameter space or an alternative elimination.
minor comments (4)
  1. [§4.2.2] The phrase 'local identifiability of (Θ,p^u)' on Ω′ is confusing because Ω′ is parameterized by (Θ,μ,Σ), not by (Θ,p^u). Clarify whether the conclusion applies to the composed map (Θ,μ,Σ) → response distribution.
  2. [§4.2.2] The paragraph beginning 'In the next subsection, we will show that the marginal probabilities...' is repeated verbatim, and the proof of real analyticity of p^u(μ,Σ) is deferred but never fully supplied in §4.3. Please either prove it in §4.2.2 or provide a precise reference.
  3. [§4.2.1] The Macaulay2 computation that rank(J(φ))=21 at (Θ0,p^u0) is central; please state explicitly that this point does lie in Ω′ (with μ0=0, Σ0=I) and include the relevant code or output in the online supplement so the computation is reproducible.
  4. [§5.1] The statement 'Because identifiability is preserved under these parameter restrictions' should be expanded: a submodel of a locally identifiable model is locally identifiable, but the measure-zero caveat in Definition 1 should be acknowledged.

Circularity Check

0 steps flagged

No significant circularity: the derivation is self-contained, built on external algebraic/analytic tools; the only overlapping-author citation is definitional (the model itself), not load-bearing for the identifiability claim.

full rationale

The paper's derivation chain does not reduce any target result to its own inputs. (1) The PM-CDM model definition (Gaussian copula on mastery scores, item responses averaged over realized binary profiles) is cited to Shang et al. (2021), which shares co-author Erosheva; however, this is the object of study, not a premise used to prove identifiability, and the paper re-derives the RLCM representation itself via Eq. (3.5). (2) Step 1 establishes full generic rank of the unconstrained Jacobian by evaluating a 21x21 minor at one explicit point (Theta0, mu0, Sigma0) using Macaulay2 code deposited at OSF, then extends nonvanishing to the copula-restricted subspace by real-analytic continuation (Krantz-Parks 2002; Okamoto 1973; Sullivant 2018 -- all external, none stating the target result). This is a witness computation for a nonzero rational/analytic function, not a fit to data and not a quantity defined in terms of the conclusion. (3) Step 2 identifies (mu,Sigma) from reduced class probabilities using explicit bivariate-normal CDF equations derived in-text, with injectivity argued from strict monotonicity; no fitted parameter is renamed as a prediction. (4) Step 3 identifies remaining item parameters by solving linear systems whose coefficient Gram matrix is proved positive definite in-text. The cited propositions from Xu (2017), Sullivant (2018), and Krantz-Parks (2002) are external, published, and their assumptions do not include the target result. A proof-technical gap may exist in transferring full Jacobian rank to finite fibers (a nonzero ambient 21x21 minor vs. the 15-parameter restricted Jacobian; generic submersion vs. finite preimages), but that is a correctness/rigor concern, not circularity: the paper does not assume what it proves. The paper also honestly states its limitation that only generic finite-to-one, not global one-to-one, identifiability is established. Hence no circular step can be exhibited; score 1 reflects only the minor, non-load-bearing self-citation for the model definition.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No numbers are fitted to data and no new entities are postulated. The central claim rests on the PM-CDM model definition, standard algebraic-statistics theorems, real-analyticity of Gaussian integrals, and one unproved ad hoc transfer assumption about the exceptional set. The inconsistency about uniform marginals is a modeling assumption that is not reflected in the parameterization actually used.

axioms (5)
  • domain assumption PM-CDM model: z=Φ^{-1}(d) ~ N(μ,Σ), d_k=Φ(z_k), with item-specific Bernoulli draws.
    Section 2.2 defines the model. The paper also states each d_k is uniform, which only holds if μ_k=0 and σ_kk=1; the proof nonetheless treats (μ,Σ) as free. This inconsistency is never resolved.
  • standard math Algebraic statistics theorem: full generic Jacobian rank of the unconstrained (Θ,p_u) parameterization implies generic finite-to-one identifiability on the full domain.
    Used in Section 4.2.1 via Sullivant Prop 16.1.7 and Okamoto's lemma to go from rank 21 at one point to generically finite fibers on Ω.
  • standard math Real-analytic zero-set lemma (Corollary 1 from Krantz-Parks) and analyticity of the Gaussian copula map (μ,Σ)→p_u.
    Used in Section 4.2.2 to transfer a.e. statements from the full domain to the lower-dimensional copula submanifold; the analyticity of Gaussian integrals is plausible but only sketched.
  • ad hoc to paper The Gaussian-copula image avoids the exceptional algebraic set where the unconstrained map has positive-dimensional fibers; equivalently, Jacobian full rank a.e. on Ω′ implies finite fibers a.e. on Ω′.
    This is the unproved load-bearing step. The paper proves avoidance of the rank-deficient set V, but algebraic maps can have infinite fibers at full-rank points (e.g., the fiber of (x(x-1/2),xy) over (0,0) contains the line x=0 while the isolated point x=1/2 has full rank). The manuscript does not control the exceptional set.
  • standard math Xu's T-matrix transformation result (Prop 1/3 in Xu 2017) used to build linear systems in Step 3.
    Used in Section 4.4 to isolate the unknown item parameters θ_{j,β}; this is an external, established result with independent support.

pith-pipeline@v1.3.0-alltime-deepseek · 20189 in / 47366 out tokens · 456271 ms · 2026-08-01T20:33:39.688562+00:00 · methodology

0 comments
read the original abstract

Partial-mastery (PM) cognitive diagnostic models (CDMs) extend traditional CDMs by replacing binary latent attribute mastery indicators with continuous mastery scores for multiple latent attributes. In PM-CDMs, each subject is characterized by a fixed continuous latent mastery vector, from which item-specific binary attribute profiles are independently generated. This formulation provides a bridge between classical CDMs and continuous latent variable models. Despite growing interest in PM-CDMs, their identifiability properties remain unexplored. In this work, we establish the first identifiability results for PM-CDMs. We derive sufficient conditions for identifiability that are direct analogues of established conditions for traditional CDMs. To develop the main argument, we use symbolic computation on a minimal example with five items and two latent attributes to show that the Jacobian of the model parameterization is generically nonzero. Combining tools from real analysis and algebraic statistics, we prove that this local property implies generic finite-to-one identifiability of the item parameters and the marginal distributions of the relevant latent attributes. We further show that if the $Q$-matrix contains such identifiable local structures for all attribute pairs, identifiability extends to the full PM-CDM. These findings provide a rigorous theoretical foundation for estimation and inference in partial-mastery cognitive diagnostic models.

Figures

Figures reproduced from arXiv: 2607.16593 by Elena Erosheva, Jun Wu, Patr\'icia Martinkov\'a.

Figure 1
Figure 1. Figure 1: Overview of the proof strategy for Theorem 1. Step 1 establishes local identifiability of a reduced submodel, Step 2 identifies the copula parameters (µ, Σ), and Step 3 recovers the remaining item parameters. 4.2 Step 1: A Submodel with Full-rank Jacobian In this step, we establish local identifiability of a specific submodel via the Jacobian criterion. 4.2.1 Reparameterization of p in a Submodel For items… view at source ↗

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Works this paper leans on

111 extracted references · 6 linked inside Pith

  1. [1]

    The Annals of Statistics , volume=

    Identifiability of restricted latent class models with binary responses , author=. The Annals of Statistics , volume=. 2017 , publisher=

    2017

  2. [2]

    Psychometrika , volume=

    The sufficient and necessary condition for the identifiability and estimability of the DINA model , author=. Psychometrika , volume=. 2019 , publisher=

    2019

  3. [3]

    The Annals of Statistics , volume=

    IDENTIFIABILITY OF PARAMETERS IN LATENT STRUCTURE MODELS WITH MANY OBSERVED VARIABLES , author=. The Annals of Statistics , volume=

  4. [4]

    The Annals of Statistics , volume=

    PARTIAL IDENTIFIABILITY OF RESTRICTED LATENT CLASS MODELS , author=. The Annals of Statistics , volume=. 2020 , publisher=

    2020

  5. [5]

    Psychometrika , pages=

    Sufficient and Necessary Conditions for the Identifiability of DINA Models with Polytomous Responses , author=. Psychometrika , pages=. 2024 , publisher=

    2024

  6. [6]

    psychometrika , volume=

    Generic identifiability of the DINA model and blessing of latent dependence , author=. psychometrika , volume=. 2023 , publisher=

    2023

  7. [7]

    Statistica Sinica , volume=

    Sufficient and necessary conditions for the identifiability of the Q-matrix , author=. Statistica Sinica , volume=. 2021 , publisher=

    2021

  8. [8]

    European Conference on Symbolic and Quantitative Approaches with Uncertainty , pages=

    On Identifiability of BN2A Networks , author=. European Conference on Symbolic and Quantitative Approaches with Uncertainty , pages=. 2023 , organization=

    2023

  9. [9]

    arXiv preprint arXiv:2505.18410 , year=

    Identifiability of latent causal graphical models without pure children , author=. arXiv preprint arXiv:2505.18410 , year=

    arXiv

  10. [10]

    psychometrika , volume=

    A restricted four-parameter IRT model: The Dyad four-parameter Normal Ogive (Dyad-4PNO) model , author=. psychometrika , volume=. 2020 , publisher=

    2020

  11. [11]

    Sparse Bayesian Multidimensional Item Response Theory , ISSN=

    Li, Jiguang and Gibbons, Robert and Ročková, Veronika , year=. Sparse Bayesian Multidimensional Item Response Theory , ISSN=. doi:10.1080/01621459.2025.2476786 , journal=

    arXiv 2025

  12. [12]

    Statistica Sinica , volume=

    Identifiability of bifactor models , author=. Statistica Sinica , volume=. 2021 , publisher=

    2021

  13. [13]

    Proceedings of the third Berkeley symposium on mathematical statistics and probability , volume=

    Statistical inference in factor analysis , author=. Proceedings of the third Berkeley symposium on mathematical statistics and probability , volume=

  14. [14]

    Econometrics , volume=

    When it counts—Econometric identification of the basic factor model based on glt structures , author=. Econometrics , volume=. 2023 , publisher=

    2023

  15. [15]

    Linear Algebra and its Applications , volume=

    Identifiability of factor analysis: Some results and open problems , author=. Linear Algebra and its Applications , volume=. 1985 , publisher=

    1985

  16. [16]

    Bernoulli: official journal of the Bernoulli Society for Mathematical Statistics and Probability , volume=

    Theory of the self-learning Q-matrix , author=. Bernoulli: official journal of the Bernoulli Society for Mathematical Statistics and Probability , volume=. 2013 , publisher=

    2013

  17. [17]

    Journal of the American Statistical Association , volume=

    Statistical analysis of Q-matrix based diagnostic classification models , author=. Journal of the American Statistical Association , volume=. 2015 , publisher=

    2015

  18. [18]

    psychometrika , volume=

    Learning latent and hierarchical structures in cognitive diagnosis models , author=. psychometrika , volume=. 2023 , publisher=

    2023

  19. [19]

    arXiv preprint arXiv:2402.13961 , year=

    New directions in algebraic statistics: Three challenges from 2023 , author=. arXiv preprint arXiv:2402.13961 , year=

    Pith/arXiv arXiv 2023

  20. [20]

    Psychometrika , volume=

    Going Deep in Diagnostic Modeling: Deep Cognitive Diagnostic Models (DeepCDMs) , author=. Psychometrika , volume=. 2024 , publisher=

    2024

  21. [21]

    Bernoulli , year=

    Blessing of Dependence: Identifiability and Geometry of Discrete Models with Multiple Binary Latent Variables , author=. Bernoulli , year=

  22. [22]

    Psychometrika , year=

    New Paradigm of Identifiable General-response Cognitive Diagnostic Models: Beyond Categorical Data , author=. Psychometrika , year=

  23. [23]

    The Annals of Applied Statistics , volume=

    Latent conjunctive Bayesian network: Unify attribute hierarchy and Bayesian network for cognitive diagnosis , author=. The Annals of Applied Statistics , volume=. 2024 , publisher=

    2024

  24. [24]

    2010 , publisher=

    Diagnostic Measurement: Theory, Methods, and Applications , author=. 2010 , publisher=

    2010

  25. [25]

    , author=

    Measurement of psychological disorders using cognitive diagnosis models. , author=. Psychological methods , volume=. 2006 , publisher=

    2006

  26. [26]

    Psychometrika , volume=

    The generalized DINA model framework , author=. Psychometrika , volume=. 2011 , publisher=

    2011

  27. [27]

    Proceedings of the 1977 computerized adaptive testing conference , volume=

    A model for testing with multidimensional items , author=. Proceedings of the 1977 computerized adaptive testing conference , volume=

    1977

  28. [28]

    Journal of educational and behavioral statistics , volume=

    Teacher’s corner: origin of the scaling constant d= 1.7 in item response theory , author=. Journal of educational and behavioral statistics , volume=. 1994 , publisher=

    1994

  29. [29]

    1953 , publisher=

    Estimation of the dosage mortality relationship when the dose is subject to error , author=. 1953 , publisher=

    1953

  30. [30]

    Handbook of statistics , volume=

    18 Multidimensional item response theory , author=. Handbook of statistics , volume=. 2006 , publisher=

    2006

  31. [31]

    Multidimensional Item Response Theory , pages=

    Multidimensional Item Response Theory Models , author=. Multidimensional Item Response Theory , pages=. 2009 , publisher=

    2009

  32. [32]

    The Annals of Applied Statistics , volume=

    Partial-mastery cognitive diagnosis models , author=. The Annals of Applied Statistics , volume=. 2021 , publisher=

    2021

  33. [33]

    The Annals of Statistics , pages=

    STATISTICAL ANALYSIS OF FACTOR MODELS OF HIGH DIMENSION , author=. The Annals of Statistics , pages=. 2012 , publisher=

    2012

  34. [34]

    Journal of the American Statistical Association , volume=

    Bayesian inference for general Gaussian graphical models with application to multivariate lattice data , author=. Journal of the American Statistical Association , volume=. 2011 , publisher=

    2011

  35. [35]

    2018 , publisher=

    Algebraic statistics , author=. 2018 , publisher=

    2018

  36. [36]

    2008 , publisher=

    Lectures on algebraic statistics , author=. 2008 , publisher=

    2008

  37. [37]

    The Annals of Statistics , pages=

    Distinctness of the eigenvalues of a quadratic form in a multivariate sample , author=. The Annals of Statistics , pages=. 1973 , publisher=

    1973

  38. [38]

    Computational aspects of psychometric methods: With R , author=

  39. [39]

    2020 15th Conference on Computer Science and Information Systems (FedCSIS) , pages=

    Feasibility of computerized adaptive testing evaluated by Monte-Carlo and post-hoc simulations , author=. 2020 15th Conference on Computer Science and Information Systems (FedCSIS) , pages=. 2020 , organization=

    2020

  40. [40]

    Learning and Instruction , volume=

    Is academic tracking related to gains in learning competence? Using propensity score matching and differential item change functioning analysis for better understanding of tracking implications , author=. Learning and Instruction , volume=. 2020 , publisher=

    2020

  41. [41]

    Journal of Computer Assisted Learning , volume=

    Can video games change attitudes towards history? Results from a laboratory experiment measuring short-and long-term effects , author=. Journal of Computer Assisted Learning , volume=. 2021 , publisher=

    2021

  42. [42]

    Informatica , volume=

    Learning the structure of Bayesian networks from incomplete data using a mixture model , author=. Informatica , volume=

  43. [43]

    Prevention science , volume=

    Latent class analysis: an alternative perspective on subgroup analysis in prevention and treatment , author=. Prevention science , volume=. 2013 , publisher=

    2013

  44. [44]

    PARTIAL-MASTERY COGNITIVE DIAGNOSIS MODELS

    SUPPLEMENTARY MATERIAL FOR “PARTIAL-MASTERY COGNITIVE DIAGNOSIS MODELS” , author=

  45. [45]

    Conference on Uncertainty in Artificial Intelligence , pages=

    Structure learning for cyclic linear causal models , author=. Conference on Uncertainty in Artificial Intelligence , pages=. 2020 , organization=

    2020

  46. [46]

    arXiv preprint arXiv:2308.01821 , year=

    Identifiability of Homoscedastic Linear Structural Equation Models using Algebraic Matroids , author=. arXiv preprint arXiv:2308.01821 , year=

    Pith/arXiv arXiv

  47. [47]

    Advances in Applied Mathematics , volume=

    Identifiability of homoscedastic linear structural equation models using algebraic matroids , author=. Advances in Applied Mathematics , volume=. 2025 , publisher=

    2025

  48. [48]

    IEEE Journal on Selected Areas in Information Theory , pages=

    Partial homoscedasticity in causal discovery with linear models , author=. IEEE Journal on Selected Areas in Information Theory , pages=. 2023 , publisher=

    2023

  49. [49]

    2018 , publisher=

    Handbook of graphical models , author=. 2018 , publisher=

    2018

  50. [50]

    The Annals of Statistics , pages=

    On chain graph models for description of conditional independence structures , author=. The Annals of Statistics , pages=. 1998 , publisher=

    1998

  51. [51]

    2006 , publisher=

    Probabilistic conditional independence structures , author=. 2006 , publisher=

    2006

  52. [52]

    Physics Reports , volume=

    Causality detection based on information-theoretic approaches in time series analysis , author=. Physics Reports , volume=. 2007 , publisher=

    2007

  53. [53]

    Nature communications , volume=

    Identifying causal gateways and mediators in complex spatio-temporal systems , author=. Nature communications , volume=. 2015 , publisher=

    2015

  54. [54]

    Combinatorica , volume=

    Counting flags in triangle-free digraphs , author=. Combinatorica , volume=. 2017 , publisher=

    2017

  55. [55]

    Journal of the Royal Statistical Society Series A: Statistics in Society , volume=

    When zero may not be zero: A cautionary note on the use of inter-rater reliability in evaluating grant peer review , author=. Journal of the Royal Statistical Society Series A: Statistics in Society , volume=. 2021 , publisher=

    2021

  56. [56]

    Computers and biomedical research , volume=

    Mathematical typology: a grade of membership technique for obtaining disease definition , author=. Computers and biomedical research , volume=. 1978 , publisher=

    1978

  57. [57]

    2015 , publisher=

    Handbook of mixed membership models and their applications , author=. 2015 , publisher=

    2015

  58. [58]

    Proceedings of the National Academy of Sciences , volume=

    Mixed-membership models of scientific publications , author=. Proceedings of the National Academy of Sciences , volume=. 2004 , publisher=

    2004

  59. [59]

    The Annals of Applied Statistics , volume=

    Describing disability through individual-level mixture models for multivariate binary data , author=. The Annals of Applied Statistics , volume=. 2007 , publisher=

    2007

  60. [60]

    psychometrika , volume=

    A spectral method for identifiable grade of membership analysis with binary responses , author=. psychometrika , volume=. 2024 , publisher=

    2024

  61. [61]

    Journal of machine learning research: JMLR , volume=

    Dimension-Grouped Mixed Membership Models for Multivariate Categorical Data , author=. Journal of machine learning research: JMLR , volume=

  62. [62]

    Annales de la Facult

    Alternative statistical models and representations for large sparse multi-dimensional contingency tables , author=. Annales de la Facult

  63. [63]

    Psychometrika , volume=

    Comparing latent structures of the grade of membership, Rasch, and latent class models , author=. Psychometrika , volume=. 2005 , publisher=

    2005

  64. [64]

    2021 , author =

    Identifiability in phylogenetics using algebraic matroids , journal =. 2021 , author =

    2021

  65. [65]

    The 50th anniversary of

    Drton, Mathias , TITLE =. The 50th anniversary of. 2018 , MRCLASS =

    2018

  66. [66]

    2014 , eprint=

    Computing Algebraic Matroids , author=. 2014 , eprint=

    2014

  67. [67]

    2020 , url=

    Identifiability in Phylogenetics Using Algebraic Matroids , author=. 2020 , url=

    2020

  68. [68]

    Biometrika , volume =

    Chen, Wenyu and Drton, Mathias and Wang, Y Samuel , title = ". Biometrika , volume =. 2019 , month =

    2019

  69. [69]

    Identifiability of Gaussian structural equation models with equal error variances , volume =

    Jonas Peters and Peter Bühlmann , journal =. Identifiability of Gaussian structural equation models with equal error variances , volume =

  70. [70]

    International Conference on Machine Learning , pages=

    Characterizing distribution equivalence and structure learning for cyclic and acyclic directed graphs , author=. International Conference on Machine Learning , pages=. 2020 , organization=

    2020

  71. [71]

    Probabilistic and Causal Inference , year=

    Equivalence and Synthesis of Causal Models , author=. Probabilistic and Causal Inference , year=

  72. [72]

    Journal of Computational and Graphical Statistics , volume=

    Implementations of the Monte Carlo EM algorithm , author=. Journal of Computational and Graphical Statistics , volume=. 2001 , publisher=

    2001

  73. [73]

    The annals of mathematical statistics , pages=

    A stochastic approximation method , author=. The annals of mathematical statistics , pages=. 1951 , publisher=

    1951

  74. [74]

    Journal of Educational and Behavioral Statistics , volume=

    Metropolis-Hastings Robbins-Monro algorithm for confirmatory item factor analysis , author=. Journal of Educational and Behavioral Statistics , volume=. 2010 , publisher=

    2010

  75. [75]

    psychometrika , volume=

    Computation for latent variable model estimation: A unified stochastic proximal framework , author=. psychometrika , volume=. 2022 , publisher=

    2022

  76. [76]

    psychometrika , volume=

    A mixed stochastic approximation EM (MSAEM) algorithm for the estimation of the four-parameter normal ogive model , author=. psychometrika , volume=. 2023 , publisher=

    2023

  77. [77]

    British Journal of Mathematical and Statistical Psychology , volume=

    Gaussian variational estimation for multidimensional item response theory , author=. British Journal of Mathematical and Statistical Psychology , volume=. 2021 , publisher=

    2021

  78. [78]

    Wallmark, Joakim , year =

  79. [79]

    arXiv preprint arXiv:2511.20191 , year=

    A Generalized Additive Partial-Mastery Cognitive Diagnosis Model , author=. arXiv preprint arXiv:2511.20191 , year=

    Pith/arXiv arXiv

  80. [80]

    Journal of the American statistical Association , volume=

    Variational inference: A review for statisticians , author=. Journal of the American statistical Association , volume=. 2017 , publisher=

    2017

Showing first 80 references.