Algebraic problems in structural equation modeling

Drton, Mathias

doi:10.2969/aspm/07710035

Cited by 22 publications

(32 citation statements)

References 43 publications

(73 reference statements)

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“…Ho & Nguyen (2016) proposed a general algebraic statistics framework for singular finite-mixture models, and showed that the optimal convergence rate for skewednormal mixtures is n −1/12 . More generally, singular learning theory is studied in (Watanabe, 2009(Watanabe, , 2013, and the algebraic structures of Gaussian mixture/graphical models and structural equation models are explored in (Leung et al, 2016;Drton et al, 2011;Drton, 2016).…”

Section: Related Workmentioning

confidence: 99%

Convergence rates of latent topic models under relaxed identifiability conditions

Wang

2019

Electron. J. Statist.

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In this paper we study the frequentist convergence rate for the Latent Dirichlet Allocation (Blei et al., 2003) topic models. We show that the maximum likelihood estimator converges to one of the finitely many equivalent parameters in Wasserstein's distance metric at a rate of n −1/4 without assuming separability or non-degeneracy of the underlying topics and/or the existence of more than three words per document, thus generalizing the previous works of Anandkumar et al. (2012Anandkumar et al. ( , 2014 from an information-theoretical perspective. We also show that the n −1/4 convergence rate is optimal in the worst case.

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Section: Related Workmentioning

confidence: 99%

Convergence rates of latent topic models under relaxed identifiability conditions

Wang

2019

Electron. J. Statist.

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“…Second, even though spectral methods are often less refined, the graph-theoretic approach cannot easily address parameter-dependent behavior; in particular, the unique effect of the SNR we observed in our simulations (and noted in (Balakrishnan et al 2017)). Currently, there is very little we are able to say about the impact of the SNR, and a combined approach might be more fruitful (Drton 2018).…”

Section: Discussionmentioning

confidence: 99%

Random Intersection Graphs and Missing Data

Salti

Berchenko

2020

AAAI

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Random-graphs and statistical inference with missing data are two separate topics that have been widely explored each in its field. In this paper we demonstrate the relationship between these two different topics and take a novel view of the data matrix as a random intersection graph. We use graph properties and theoretical results from random-graph theory, such as connectivity and the emergence of the giant component, to identify two threshold phenomena in statistical inference with missing data: loss of identifiability and slower convergence of algorithms that are pertinent to statistical inference such as expectation-maximization (EM). We provide two examples corresponding to these threshold phenomena and illustrate the theoretical predictions with simulations that are consistent with our reduction.

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“…, p}, i ∈ J and j ∈ J, let β i|J = (Σ iJ Σ −1 JJ ) t denote the vector of regression coefficients of X i on X J , with its j-th entry being β ij|J , the coefficient corresponding to variable X j . We may equivalently write the statistical model for X as a system of recursive linear regression equations (also called a linear structural equation model [22]), where for each i ∈ {1, . .…”

Section: A the Inverse Covariance Matrix And Systems Of Recursive Rementioning

confidence: 99%

Sparse Cholesky Covariance Parametrization for Recovering Latent Structure in Ordered Data

et al. 2020

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The sparse Cholesky parametrization of the inverse covariance matrix is directly related to Gaussian Bayesian networks. Its counterpart, the covariance Cholesky factorization model, has a natural interpretation as a hidden variable model for ordered signal data. Despite this, it has received little attention so far, with few notable exceptions. To fill this gap, in this paper we focus on arbitrary zero patterns in the Cholesky factor of a covariance matrix. We discuss how these models can also be extended, in analogy with Gaussian Bayesian networks, to data where no apparent order is available. For the ordered scenario, we propose a novel estimation method that is based on matrix loss penalization, as opposed to the existing regression-based approaches. The performance of this sparse model for the Cholesky factor, together with our novel estimator, is assessed in a simulation setting , as well as over spatial and temporal real data where a natural ordering arises among the variables. We give guidelines, based on the empirical results, about which of the methods analysed is more appropriate for each setting.

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Algebraic problems in structural equation modeling

Cited by 22 publications

References 43 publications

Convergence rates of latent topic models under relaxed identifiability conditions

Convergence rates of latent topic models under relaxed identifiability conditions

Random Intersection Graphs and Missing Data

Sparse Cholesky Covariance Parametrization for Recovering Latent Structure in Ordered Data

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