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**Nested covariance determinants and restricted trek separation in Gaussian graphical models.**
*(English)*
Zbl 1476.62119

This paper is placed in the context of Gaussian graphical models [S. L. Lauritzen, Graphical models. Oxford: Oxford Univ. Press (1996; Zbl 0907.62001)], and its focus is on acyclic directed mixed graphs (ADMGs), which are an extension of the better known directed acylic graphs (DAGs). Specifically, the authors are interested in explaining the relations among the entries of a covariance matrix that belongs to such linear structural equation models.

It is well known in the graphical models literature that probabilistic conditional independence (CI) statements among the entries of Gaussian random vectors correspond to the vanishing of specific minors of the covariance matrix, i.e., a CI statement holds if and only if the determinant of certain submatrix of the covariance matrix is zero. These are enough to completely describe the constraints in the covariance matrix of a DAG model, and correspond graphically to d-separation [M. Studený, Probabilistic conditional independence structures. London: Springer (2005; Zbl 1070.62001)]. However, in the presence of hidden variables (represented in the mixed graph by bidirected edges between observed nodes) this is no longer the case. In particular, there may be minors that vanish on the model that do not correspond to CI statements.

The work of [S. Sullivant et al., Ann. Stat. 38, No. 3, 1665–1685 (2010; Zbl 1189.62091)] introduced the concept of trek separation and managed to characterize all vanishing minors of the covariance matrix. However, there may exist other polynomial constraints that hold for ADMG models that are not of this form, such as the (in)famous Verma constraint [T. van Ommen and J. M. Mooij, “Algebraic equivalence of linear structural equation models”, in: Proceedings of the 33rd Annual Conference on Uncertainty in Artificial Intelligence (UAI-17) (2017), arXiv:1807.03527]. These kinds of relations appear to be harder to understand, and the main contribution of the paper under review is that while they cannot be expressed as minors of the covariance matrix, they can actually be expressed as determinants of matrices whose entries are determinants themselves: nested determinants. Furthermore, in their main theorem the authors are able to explain the structure of many of these instances via the concept of restricted trek-separation.

Through several illuminating and well-chosen examples, the authors illustrate not only how the theory they develop applies to different kinds of ADMGs, but also how there are other graphical models, such as ones that now allow cycles, which have constraints that also admit a nested determinant representation (which is in general not unique). Indeed, these observations reveal that the story of nested determinants and graphical model relations is not fully understood yet, and the paper suggests many interesting open problems for follow-up work in this direction.

It is well known in the graphical models literature that probabilistic conditional independence (CI) statements among the entries of Gaussian random vectors correspond to the vanishing of specific minors of the covariance matrix, i.e., a CI statement holds if and only if the determinant of certain submatrix of the covariance matrix is zero. These are enough to completely describe the constraints in the covariance matrix of a DAG model, and correspond graphically to d-separation [M. Studený, Probabilistic conditional independence structures. London: Springer (2005; Zbl 1070.62001)]. However, in the presence of hidden variables (represented in the mixed graph by bidirected edges between observed nodes) this is no longer the case. In particular, there may be minors that vanish on the model that do not correspond to CI statements.

The work of [S. Sullivant et al., Ann. Stat. 38, No. 3, 1665–1685 (2010; Zbl 1189.62091)] introduced the concept of trek separation and managed to characterize all vanishing minors of the covariance matrix. However, there may exist other polynomial constraints that hold for ADMG models that are not of this form, such as the (in)famous Verma constraint [T. van Ommen and J. M. Mooij, “Algebraic equivalence of linear structural equation models”, in: Proceedings of the 33rd Annual Conference on Uncertainty in Artificial Intelligence (UAI-17) (2017), arXiv:1807.03527]. These kinds of relations appear to be harder to understand, and the main contribution of the paper under review is that while they cannot be expressed as minors of the covariance matrix, they can actually be expressed as determinants of matrices whose entries are determinants themselves: nested determinants. Furthermore, in their main theorem the authors are able to explain the structure of many of these instances via the concept of restricted trek-separation.

Through several illuminating and well-chosen examples, the authors illustrate not only how the theory they develop applies to different kinds of ADMGs, but also how there are other graphical models, such as ones that now allow cycles, which have constraints that also admit a nested determinant representation (which is in general not unique). Indeed, these observations reveal that the story of nested determinants and graphical model relations is not fully understood yet, and the paper suggests many interesting open problems for follow-up work in this direction.

Reviewer: Carlos Améndola (München)

### MSC:

62H22 | Probabilistic graphical models |

62H12 | Estimation in multivariate analysis |

62J10 | Analysis of variance and covariance (ANOVA) |

62R01 | Algebraic statistics |

00A27 | Lists of open problems |

### Keywords:

covariance matrix; conditional independence; graphical model; trek separation; Verma constraint; open problems### Software:

TETRAD### References:

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