Polynomial systems of graphical models Elizabeth Gross University - PowerPoint PPT Presentation

Polynomial systems of graphical models Elizabeth Gross University of Hawai‘i at M¯ anoa ICERM Nonlinear Algebra, Graphical Models Working Group: Bibhas Adhikari, Alexandros Grosdos, Marc H¨ ark¨ onen, Cvetelina Hill, Sara Lamboglia, Samantha Sherman, Elias Tsigaridas, Dane Wilburne seal.jpg Elizabeth Gross, UH M¯ anoa Structural Equation Models

Graphical Models Gaussian graphical models Undirected graphs Described in detail in ”Gaussian Graphical Models: An Algebraic and Geometric Perspective”, Uhler (2017) Conjecture on ML Degree of cycles in Section 7.4, Lectures on Algebraic Statistics , Drton–Sturfmels–Sullivant (2009) Linear Structural Equation Models Directed graphs, mixed graphs Overview in ”Algebraic Problems in Structural Equation Modeling”, Drton (2018) Elizabeth Gross, UH M¯ anoa Structural Equation Models

Polynomial systems of graphical models Elizabeth Gross University of Hawai‘i at M¯ anoa ICERM Nonlinear Algebra, Graphical Models Working Group: Bibhas Adhikari, Alexandros Grosdos, Marc H¨ ark¨ onen, Cvetelina Hill, Sara Lamboglia, Samantha Sherman, Elias Tsigaridas, Dane Wilburne seal.jpg Elizabeth Gross, UH M¯ anoa Structural Equation Models

Polynomial systems of graphical models Elizabeth Gross University of Hawai‘i at M¯ anoa ICERM Nonlinear Algebra, Graphical Models Working Group: Bibhas Adhikari, Alexandros Grosdos, Marc H¨ ark¨ onen, Cvetelina Hill, Sara Lamboglia, Samantha Sherman, Elias Tsigaridas, Dane Wilburne seal.jpg Elizabeth Gross, UH M¯ anoa Structural Equation Models

Structural Equation Models R D = Λ 2 R V × V : λ ij = 0 if i ! j / � 2 D R D reg = subset of matrices Λ 2 R D for which I � Λ is invertible. PD V = cone of pos def symmetric V ⇥ V matrices. PD ( B ) = { Ω 2 PD V : ω ij = 0 if i 6 = j and i $ j / 2 B } . Definition The linear structural equation model given by a mixed graph G = ( V , D , B ) on V = [ m ] is the family of all probability distributions on R m with covariance matrix Σ = ( I � Λ ) − T Ω ( I � Λ ) − 1 for Λ 2 R D reg and Ω 2 PD ( B ). Elizabeth Gross, UH M¯ anoa Structural Equation Models

Structural Equation Models R D = Λ 2 R V × V : λ ij = 0 if i ! j / � 2 D R D reg = subset of matrices Λ 2 R D for which I � Λ is invertible. PD V = cone of pos def symmetric V ⇥ V matrices. PD ( B ) = { Ω 2 PD V : ω ij = 0 if i 6 = j and i $ j / 2 B } . Definition The linear structural equation model given by a mixed graph G = ( V , D , B ) on V = [ m ] is the family of all probability distributions on R m with covariance matrix Σ = ( I � Λ ) − T Ω ( I � Λ ) − 1 for Λ 2 R D reg and Ω 2 PD ( B ). Elizabeth Gross, UH M¯ anoa Structural Equation Models

Identifiability The covariance parametrization is φ G : R D ⇥ PD ( B ) ! PD V ( Λ , Ω ) 7! ( I � Λ ) � T Ω ( I � Λ ) � 1 Definition The fiber of a pair ( Λ , Ω ) 2 R D reg ⇥ PD ( B ) is ( Λ 0 , Ω 0 ) 2 R D � reg ⇥ PD ( B ) : φ G ( Λ 0 , Ω 0 ) = φ G ( Λ , Ω ) F G ( Λ , Ω ) = Elizabeth Gross, UH M¯ anoa Structural Equation Models

Identifiability The covariance parametrization is φ G : R D ⇥ PD ( B ) ! PD V ( Λ , Ω ) 7! ( I � Λ ) � T Ω ( I � Λ ) � 1 Definition The fiber of a pair ( Λ , Ω ) 2 R D reg ⇥ PD ( B ) is ( Λ 0 , Ω 0 ) 2 R D � reg ⇥ PD ( B ) : φ G ( Λ 0 , Ω 0 ) = φ G ( Λ , Ω ) F G ( Λ , Ω ) = If the map φ G is injective, then we call the model global identifiable . If the map φ G is generically injective, then we call the model generically identifiable . If the map φ G is generically k -to-one, then we call the model generically locally identifiable . In this case, we call k the identifiability degree of the model. Elizabeth Gross, UH M¯ anoa Structural Equation Models

Identifiability The covariance parametrization is φ G : R D ⇥ PD ( B ) ! PD V ( Λ , Ω ) 7! ( I � Λ ) � T Ω ( I � Λ ) � 1 Definition The fiber of a pair ( Λ , Ω ) 2 R D reg ⇥ PD ( B ) is ( Λ 0 , Ω 0 ) 2 R D � reg ⇥ PD ( B ) : φ G ( Λ 0 , Ω 0 ) = φ G ( Λ , Ω ) F G ( Λ , Ω ) = If the map φ G is injective, then we call the model global identifiable . If the map φ G is generically injective, then we call the model generically identifiable . If the map φ G is generically k -to-one, then we call the model generically locally identifiable . In this case, we call k the identifiability degree of the model. Elizabeth Gross, UH M¯ anoa Structural Equation Models

Another set of equations Lemma (Foygel–Draisma–Drton (2012), Drton (2018)) Let G = ( V , D , B ) be a mixed graph, and let Σ = φ G ( Λ 0 , Ω 0 ) for Λ 0 2 R D reg and Ω 0 2 PD ( B ) . Then the fiber F G ( Λ 0 , Ω 0 ) is isomorphic to the set of matrices Λ 2 R D reg that solve the equation system: F ij = [( I � Λ ) T Σ ( I � Λ )] ij = 0 i 6 = j , i $ j / 2 B or more explicitly: X X X X F ij = σ ij � λ lj σ il + λ ki σ kl λ lj = 0 λ ki σ kj � k → i l → j k → i l → j Need to be careful about spurious solutions (det( I � Λ ) = 0). Elizabeth Gross, UH M¯ anoa Structural Equation Models

Another set of equations Lemma (Foygel–Draisma–Drton (2012), Drton (2018)) Let G = ( V , D , B ) be a mixed graph, and let Σ = φ G ( Λ 0 , Ω 0 ) for Λ 0 2 R D reg and Ω 0 2 PD ( B ) . Then the fiber F G ( Λ 0 , Ω 0 ) is isomorphic to the set of matrices Λ 2 R D reg that solve the equation system: F ij = [( I � Λ ) T Σ ( I � Λ )] ij = 0 i 6 = j , i $ j / 2 B or more explicitly: X X X X F ij = σ ij � λ lj σ il + λ ki σ kl λ lj = 0 λ ki σ kj � k → i l → j k → i l → j Need to be careful about spurious solutions (det( I � Λ ) = 0). Elizabeth Gross, UH M¯ anoa Structural Equation Models

Another set of equations Lemma (Foygel–Draisma–Drton (2012), Drton (2018)) Let G = ( V , D , B ) be a mixed graph, and let Σ = φ G ( Λ 0 , Ω 0 ) for Λ 0 2 R D reg and Ω 0 2 PD ( B ) . Then the fiber F G ( Λ 0 , Ω 0 ) is isomorphic to the set of matrices Λ 2 R D reg that solve the equation system: F ij = [( I � Λ ) T Σ ( I � Λ )] ij = 0 i 6 = j , i $ j / 2 B or more explicitly: X X X X F ij = σ ij � λ lj σ il + λ ki σ kl λ lj = 0 λ ki σ kj � k → i l → j k → i l → j Need to be careful about spurious solutions (det( I � Λ ) = 0). Elizabeth Gross, UH M¯ anoa Structural Equation Models

Another set of equations Lemma (Foygel–Draisma–Drton (2012), Drton (2018)) Let G = ( V , D , B ) be a mixed graph, and let Σ = φ G ( Λ 0 , Ω 0 ) for Λ 0 2 R D reg and Ω 0 2 PD ( B ) . Then the fiber F G ( Λ 0 , Ω 0 ) is isomorphic to the set of matrices Λ 2 R D reg that solve the equation system: F ij = [( I � Λ ) T Σ ( I � Λ )] ij = 0 i 6 = j , i $ j / 2 B or more explicitly: X X X X F ij = σ ij � λ lj σ il + λ ki σ kl λ lj = 0 λ ki σ kj � k → i l → j k → i l → j Need to be careful about spurious solutions (det( I � Λ ) = 0). Elizabeth Gross, UH M¯ anoa Structural Equation Models

Identifiability Results Theorem Theorem (Brito–Pearl) Acyclic and no bidirected edges G criterion ⇒ generically ⇒ globally identifiable. identifiable. Theorem Theorem (Foygel–Draisma–Drton) (Drton–Foygel–Sullivant) G does not contain a subgraph Half-trek criterion ⇒ generically identifiable. whose B is connected and D has a unique sink ⇒ globally Extended by: Chen (2015), identifiable. Drton–Weihs (2016), and Weihs–Robinson–Dufresne– Theorem (Brito–Pearl) Kenkel–Kubjas–McGee–Nguyen– Acyclic and simple ⇒ generically Robeva identifiable (2018) Elizabeth Gross, UH M¯ anoa Structural Equation Models

Identifiability Degree Results Theorem (ICERM Group) Simple ) generically locally identifiable (finite-to-one). Theorem (Drton–Foygel–Sullivant) The identifiability degree of a cycle with length � 3 is 2. Theorem (Foygel–Draisma–Drton) Analysis of the identifiability degree of mixed graphs with up to five nodes. The maximum identifiability degree observed was 10. Elizabeth Gross, UH M¯ anoa Structural Equation Models

Identifiability Degree Results Theorem (ICERM Group) Simple ) generically locally identifiable (finite-to-one). Theorem (Drton–Foygel–Sullivant) The identifiability degree of a cycle with length � 3 is 2. Theorem (Foygel–Draisma–Drton) Analysis of the identifiability degree of mixed graphs with up to five nodes. The maximum identifiability degree observed was 10. Elizabeth Gross, UH M¯ anoa Structural Equation Models

Identifiability Degree Results Theorem (ICERM Group) Simple ) generically locally identifiable (finite-to-one). Theorem (Drton–Foygel–Sullivant) The identifiability degree of a cycle with length � 3 is 2. Theorem (Foygel–Draisma–Drton) Analysis of the identifiability degree of mixed graphs with up to five nodes. The maximum identifiability degree observed was 10. Elizabeth Gross, UH M¯ anoa Structural Equation Models

Tian’s Decomposition Theorem (Tian (2005)) The degree of identifiability of a mixed graph G is the product of the degrees of identifiability of its mixed components G [ C ] , C 2 C ( G ) . In particular, φ G is (generically) injective if and only if each φ G [ C ] is so, for C 2 C ( G ) . Elizabeth Gross, UH M¯ anoa Structural Equation Models