Schedule

Abstract: A colored directed acyclic graph can be used to represent the linear Gaussian DAG models that fulfill a collection of partial homoscedasticity and partial homogeneity constraints placed on the model error variances and regression coefficients according to the coloring of the vertex and edge set of the graph, respectively.  In this talk, we will discuss how some properties of classical DAG models lift to the colored setting, including methods of parameter identification, smoothness, existence of faithful distributions and model equivalence results. On the geometric side, we will also observe that a conjectured property of the conditional independence ideal of a Gaussian DAG model holds in the more general setting of colored DAG models.  This talk is based on joint work with Tobias Boege, Kaie Kubjas and Pratik Misra.

Abstract: Phylogenetics studies the evolutionary relationships among species using their molecular sequences. These relationships are represented on a phylogenetic tree or network. Modeling nucleotide or amino acid substitution along a phylogenetic tree is one of the most common approaches in phylogenetic reconstruction. One can use a general Markov model or one of its submodels given by certain substitution symmetries. If these symmetries are governed by the action of a permutation group G on the rows and columns of a transition matrix, we speak of G-equivariant models. A Markov process on a phylogenetic tree or network parametrizes a dense subset of an algebraic variety, the so-called phylogenetic variety.

During the last decade algebraic geometry has been used in phylogenetics for phylogenetic reconstruction and to establish the identifiability of parameters of complex evolutionary models (and thus guarantee model consistency). Since G-equivariant models have fewer parameters than a general Markov model, their phylogenetic varieties are defined by more equations and these are usually hard to find. We will see that we can easily derive equations for G-equivariant models from the equations of a phylogenetic variety evolving under a general Markov model.

Joint work with Jesús Fernández-Sánchez (Universitat Politècnica de Catalunya, Spain).

Abstract: Sparse estimation of precision matrices is widely used also beyond the Gaussian case, though a vanishing partial correlation does not imply a conditional independence for non-Gaussian data. In this talk I will present a novel perspective on sparse precision matrices in terms of operator selfdecomposable distributions, which appear as steady-state distributions for a class of Markov processes. This perspective relates to: (i) a causal interpretation; (ii) a graphical interpretation; (iii) algebraic constraints. Specifically, we have derived expressions for all higher order cumulants, and we have shown that a particular symmetry constraint allows for identification of causal effects from observational data. The symmetry constraint can be expressed as a rank constraint on a matrix that involves second and third order cumulants. I will discuss the constraint and show how to test it. Joint work with Jeffrey Adams and Cecilie Olesen Recke

Abstract: Understanding evolutionary relationships, particularly in the context of hybridization and horizontal gene transfer, requires the inference of phylogenetic networks rather than traditional trees. Recent methods based on quartet concordance factors have addressed the challenges of inferring such networks, where these factors represent the probabilities of 4-taxon relationships within gene trees. Previous research has shown that certain network topologies and numerical parameters can be identified, but gaps remain in understanding the full topology of level-1 phylogenetic networks under the network multispecies coalescent model. In this talk, we investigate what level-1 network features are identifiable by studying the ideals defined by quartet concordance factors. 

We address both the identifiability of the network, as well as the numerical parameters, and will give answers to a number of identifiability problems related to the 3-cycles of the network.

Joint work with Elizabeth S. Allman, Hector Baños and John A. Rhodes

Abstract: Instrumental variables are a popular tool to estimate causal effects in the presence of unmeasured confounding. We consider the question of how to select statistically attractive conditional instrumental sets using knowledge of the underlying causal structure in the form of a graph.  We do so in the setting of a linear structural equation model with correlated errors that is compatible with a known acyclic directed mixed graph. We first characterize the class of conditional instrumental sets that yield consistent two-stage least squares estimators for the target total effect. Based on this result, we characterize a linearly valid set that has the smallest asymptotic variance among all linearly valid conditional instrumental sets that can be ensured with a graphical criterion alone.

Abstract: Graphical models in extremes permit a natural interpretation of extremal conditional independence structures. The recently introduced notion of undirected graphical models for extremes from threshold exceedances relies on a new notion of extremal conditional independence for these models.

In this talk, we propose directed graphical models for extremes from threshold exceedances and show that these models satisfy directed extremal Markov properties. Furthermore, we discuss how to construct such models via extremal structural equation models.

For the subclass of Hüsler--Reiss distributions, which are considered as an analogue of Gaussians in extremes, extremal conditional independence can be described parametrically.

Here we find that linear structural assignments allow flexible models for extremal conditional independence constraints. Under mild assumptions, such models are generically faithful to the underlying directed acyclic graph and permit a PC-type algorithm via Gaussian conditional independence testing.

Abstract

Abstract: The maximum likelihood degree, or ML-degree, of a statistical model is the number of complex critical points of its likelihood function for generic data. In this talk, we present recent results on the ML-degree of Brownian motion tree models. We use intersection theoretic techniques to prove that this ML-degree is $2^{n+1}-2n -3$. We also show that for general trees, the ML-degree is invariant under rerooting of the tree. This is joint work with Shelby Cox, Aida Maraj and Ikenna Nometa.

Abstract: For a given Acyclic Directed Mixed Graph, the Linear Non-Gaussian Acyclic Model (LiNGAM) postulates that each random variable is a linear function of its parents, with exogenous non-Gaussian error terms. In this context, we present a graphical criterion that is necessary and sufficient for deciding the generic identifiability of a causal effect within a fixed graph. We also provide an algorithm for testing this criterion, which operates in polynomial time relative to the size of the graph. Furthermore, when the graphical criteria are met, we demonstrate that the model parameters can be determined as the solution to an optimization problem, which can be solved using gradient methods.    

Abstract: We study the maximum likelihood degree (MLD) of centered multivariate Gaussian statistical models with homogeneous vanishing ideal. The MLD of a statistical model M counts the number of complex critical points of the log-likelihood function constrained to the Zariski closure of M, and this number is independent of the data. The maximum likelihood estimator (MLE) maps every data to the maximizer of the corresponding likelihood function on M and it is typically computed by numerically solving the constrained optimization problem. It is exactly when MLD = 1 that the MLE is a rational function, which can be viewed as a 'closed solution formula'. In a joint paper with  C. Améndola, K. Kohn, O. Marigliano, A. Seigal we establish a one-to-one correspondence between models with rational MLE (same as MLD =1) and the solutions to a nonlinear first-order partial differential equation. In a follow-up paper we then combine this correspondence with the "F-adjoined Gauss map"  to classify projective curves with rational MLE.

Abstract: We present a geometrically enhanced Markov chain Monte Carlo sampler for networks based on a discrete curvature measure defined on graphs. Specifically, we incorporate the concept of graph Forman curvature into sampling procedures on both the nodes and edges of a network explicitly, via the transition probability of the Markov chain, as well as implicitly, via the target stationary distribution, which gives a novel, curved Markov chain Monte Carlo approach to learning networks. We show that integrating curvature into the sampler results in faster convergence to a wide range of network statistics demonstrated on deterministic networks drawn from real-world data.

Abstract: Complete collineations are an important tool in algebraic geometry for studying degenerations of linear maps. The aim of this talk is to show how complete collineations can be used to resolve non-identifiability of the Maximum Likelihood Estimate (MLE) in the setting of directed acyclic Gaussian graphical (DAG) models. The MLE given a sample in a given statistical model corresponds to a choice of parameters for that model that maximises the log-likelihood function given the sample. The key idea for linking complete collineations to samples for DAG models is to consider a sample with non-identifiable MLE as a degenerate linear map, and to use a complete collineation to perturb this map so as to make it non-degenerate. I will explain how the MLE given such a perturbation is always unique, and tends to an MLE given the initial sample if one exists. This is joint work with Gergely Berczi, Philipp Reichenbach and Anna Seigal.

Abstract: Factor analysis is a statistical technique that explains correlations among observed random variables with the help of a smaller number of unobserved factors.  In traditional full factor analysis, each observed variable is influenced by every factor. However, many applications exhibit interesting sparsity patterns i.e. each observed variable only depends on a subset of the factors.  We study such sparse factor analysis models from an algebro-geometric perspective.  Under a mild condition on the sparsity pattern, we compute the dimension of the set of covariance matrices that corresponds to a given model.  Moreover, we study algebraic relations among the covariances in sparse two-factor models.  In particular, we identify cases in which a Gröbner basis for these relations can be derived via a 2-delightful term order and joins of toric edge ideals. This is a joint-work with Mathias Drton, Alex Grosdos and Nils Sturma.

• Tobias Boege: Colored Gaussian DAG models

• Kamillo Ferry: Enumerating weighted dag polyhedra

• Alex Grosdos: Algebraic Statistics of Huesler-Reiss Extremal Graphical Models

• Sarah Lumpp: Conditional Independence in Graphical Continuous Lyapunov Models

• Andrew McCormack: Rational Maximum Likelihood Estimators of Kronecker Covariance Matrices

• Janike Oldekop: Likelihood Geometry of Reflexive Polytopes

• Daniela Schkoda: Causal discovery of linear non-Gaussian causal models with unobserved confounding: A recursive method

• Maximilian Wiesmann: Algebraic Geometry of Quantum Graphical Models