arXiv subjects

Seth Sullivant

Publications and source records attributed to Seth Sullivant.

At least 19 recordsLinked to original sources

arXiv2025

Singular Learning Theory for Factor Analysis

Watanabe's singular learning theory provides a framework for asymptotic analysis of Bayesian model selection for statistical models with singularities, where traditional statistical regularity assumptions fail. Learning coefficients, also known as real log canonical thresholds, play a central role in singular learning, as they govern the asymptotic behavior of Bayesian marginal likelihood integrals in settings where the Laplace approximations used for regular statistical models are not applicable. Learning coefficients are algebraic invariants that quantify the geometric complexity of a model and reveal how the singular structure impacts the model's generalization properties. In this paper, we apply algebraic methods to study the learning coefficients of factor analysis models, which are widely used latent variable models for continuously distributed data. Our main result provides exact formulas for learning coefficients of factor analysis models. Moreover, we study the singularity types of specific factor analysis models in detail.

math.ST↗

arXiv2025

Identifiability of Large Phylogenetic Mixtures for Many Phylogenetic Model Structures

Identifiability of phylogenetic models is a necessary condition to ensure that the model parameters can be uniquely determined from data. Mixture models are phylogenetic models where the probability distributions in the model are convex combinations of distributions in simpler phylogenetic models. Mixture models are used to model heterogeneity in the substitution process in DNA sequences. While many basic phylogenetic models are known to be identifiable, mixture models in generality have only been shown to be identifiable in certain cases. We expand the main theorem of [Rhodes, Sullivant 2012] to prove identifiability of mixture models in equivariant phylogenetic models, specifically the Jukes-Cantor, Kimura 2-parameter model, Kimura 3-parameter model and the Strand Symmetric model.

q-bio.PE↗

arXiv2025

Phylogenetic network models as graphical models

The displayed tree phylogenetic network model is shown to sit as a natural submodel of the graphical model associated to a directed acyclic graph (DAG). This representation allows to derive a number of results about the displayed tree model. In particular, the concept of a local modification to a DAG model is developed and applied to the displayed tree model. As an application, some nonidentifiability issues related to the displayed tree models are highlighted as they relate to reticulation edges and stacked reticulations in the networks. We also derive rank conditions on flattenings of probability tensors for the displayed tree model, generalizing classic results for phylogenetic tree models.

q-bio.PE↗

arXiv2024

Lattice supported distributions and graphical models

For the distributions of finitely many binary random variables, we study the interaction of restrictions of the supports with conditional independence constraints. We prove a generalization of the Hammersley-Clifford theorem for distributions whose support is a natural distributive lattice: that is, any distribution which has natural lattice support and satisfies the pairwise Markov statements of a graph must factor according to the graph. We also show a connection to the Hibi ideals of lattices.

math.ST↗

arXiv2024

Marginal Independence and Partial Set Partitions

We establish a bijection between marginal independence models on $n$ random variables and split closed order ideals in the poset of partial set partitions. We also establish that every discrete marginal independence model is toric in cdf coordinates. This generalizes results of Boege, Petrovic, and Sturmfels and Drton and Richardson, and provides a unified framework for discussing marginal independence models. Additionally, we provide an axiomatic characterization of marginal independence and we show that our set of axioms are sound and complete in the set of probability distributions. This follows the work of Geiger, Paz and Pearl who provided an analogous characterization of independence for statements involving 2 sets of random variables.

math.ST↗

arXiv2024

Equidistant Circular Split Networks

Phylogenetic networks are generalizations of trees that allow for the modeling of non-tree like evolutionary processes. Split networks give a useful way to construct networks with intuitive distance structures induced from the associated split graph. We explore the polyhedral geometry of distance matrices built from circular split systems which have the added property of being equidistant. We give a characterization of the facet defining inequalities and the extreme rays of the cone of distances that arises from an equidistant network associated to any circular split network. We also explain a connection to the Chan-Robbins-Yuen polytope from geometric combinatorics.

math.CO↗

arXiv2023

The Codegree, Weak Maximum Likelihood Threshold, and the Gorenstein Property of Hierarchical Models

The codegree of a lattice polytope is the smallest integer dilate that contains a lattice point in the relative interior. The weak maximum likelihood threshold of a statistical model is the smallest number of data points for which there is a non-zero probability that the maximum likelihood estimate exists. The codegree of a marginal polytope is a lower bound on the maximum likelihood threshold of the associated log-linear model, and they are equal when the marginal polytope is normal. We prove a lower bound on the codegree in the case of hierarchical log-linear models and provide a conjectural formula for the codegree in general. As an application, we study when the marginal polytopes of hierarchical models are Gorenstein, including a classification of Gorenstein decomposable models, and a conjectural classification of Gorenstein binary hierarchical models.

math.CO↗

arXiv2022

Identifiability of linear compartmental tree models and a general formula for input-output equations

A foundational question in the theory of linear compartmental models is how to assess whether a model is structurally identifiable -- that is, whether parameter values can be inferred from noiseless data -- directly from the combinatorics of the model. Our main result completely answers this question for models (with one input and one output) in which the underlying graph is a bidirectional tree; moreover, identifiability of such models can be verified visually}. Models of this structure include two families of models often appearing in biological applications: catenary and mammillary models. Our analysis of such models is enabled by two supporting results, which are significant in their own right. One result gives the first general formula for the coefficients of input-output equations (certain equations that can be used to determine identifiability) that allows for input and output to be in distinct compartments}. In another supporting result, we prove that identifiability is preserved when a model is enlarged and altered in specific ways involving adding a new compartment with a bidirected edge to an existing compartment.

math.DS↗

arXiv2022

Directed Gaussian graphical models with toric vanishing ideals

Directed Gaussian graphical models are statistical models that use a directed acyclic graph (DAG) to represent the conditional independence structures between a set of jointly normal random variables. The DAG specifies the model through recursive factorization of the parametrization, via restricted conditional distributions. In this paper, we make an attempt to characterize the DAGs whose vanishing ideals are toric ideals. In particular, we give some combinatorial criteria to construct such DAGs from smaller DAGs which have toric vanishing ideals. An associated monomial map called the shortest trek map plays an important role in our description of toric Gaussian DAG models. For DAGs whose vanishing ideal is toric, we prove results about the generating sets of those toric ideals.

math.AC↗

arXiv2022

Structural Identifiability of Series-Parallel LCR Systems

We consider the identifiability problem for the parameters of series-parallel LCR circuit networks. We prove that for networks with only two classes of components (inductor-capacitor (LC), inductor-resistor (LR), and capacitor-resistor (RC)), the parameters are identifiable if and only if the number of non-monic coefficients of the constitutive equations equals the number of parameters. The notion of the "type" of the constitutive equations plays a key role in the identifiability of LC, LR, and RC networks. We also investigate the general series-parallel LCR circuits (with all three classes of components), and classify the types of constitutive equations that can arise, showing that there are 22 different types. However, we produce an example that shows that the basic notion of type that works to classify identifiability of two class networks is not sufficient to classify the identifiability of general series-parallel LCR circuits.

math.DS↗

arXiv2021

Markov Equivalence of Max-Linear Bayesian Networks

Max-linear Bayesian networks have emerged as highly applicable models for causal inference via extreme value data. However, conditional independence (CI) for max-linear Bayesian networks behaves differently than for classical Gaussian Bayesian networks. We establish the parallel between the two theories via tropicalization, and establish the surprising result that the Markov equivalence classes for max-linear Bayesian networks coincide with the ones obtained by regular CI. Our paper opens up many problems at the intersection of extreme value statistics, causal inference and tropical geometry.

math.ST↗

arXiv2021

Discrete Max-Linear Bayesian Networks

Discrete max-linear Bayesian networks are directed graphical models specified by the same recursive structural equations as max-linear models but with discrete innovations. When all of the random variables in the model are binary, these models are isomorphic to the conjunctive Bayesian network (CBN) models of Beerenwinkel, Eriksson, and Sturmfels. Many of the techniques used to study CBN models can be extended to discrete max-linear models and similar results can be obtained. In particular, we extend the fact that CBN models are toric varieties after linear change of coordinates to all discrete max-linear models.

math.ST↗

arXiv2020

Quasi-independence models with rational maximum likelihood estimator

We classify the two-way independence quasi-independence models (or independence models with structural zeros) that have rational maximum likelihood estimators, or MLEs. We give a necessary and sufficient condition on the bipartite graph associated to the model for the MLE to be rational. In this case, we give an explicit formula for the MLE in terms of combinatorial features of this graph. We also use the Horn uniformization to show that for general log-linear models $\mathcal{M}$ with rational MLE, any model obtained by restricting to a face of the cone of sufficient statistics of $\mathcal{M}$ also has rational MLE.

math.ST↗

arXiv2020

Gaussian graphical models with toric vanishing ideals

Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. They are widely used throughout natural sciences, computational biology and many other fields. Computing the vanishing ideal of the model gives us an implicit description of the model. In this paper, we resolve two conjectures of Sturmfels and Uhler from \cite{BS n CU}. In particular, we characterize those graphs for which the vanishing ideal of the Gaussian graphical model is generated in degree $1$ and $2$. These turn out to be the Gaussian graphical models whose ideals are toric ideals, and the resulting graphs are the $1$-clique sums of complete graphs.

math.AG↗

arXiv2020

The Cavender-Farris-Neyman Model with a Molecular Clock

We give a combinatorial description of the toric ideal of invariants of the Cavender-Farris-Neyman model with a molecular clock (CFN-MC) on a rooted binary phylogenetic tree and prove results about the polytope associated to this toric ideal. Key results about the polyhedral structure include that the number of vertices of this polytope is a Fibonacci number, the facets of the polytope can be described using the combinatorial "cluster" structure of the underlying rooted tree, and the volume is equal to an Euler zig-zag number. The toric ideal of invariants of the CFN-MC model has a quadratic Groebner basis with squarefree initial terms. Finally, we show that the Ehrhart polynomial of these polytopes, and therefore the Hilbert series of the ideals, depends only on the number of leaves of the underlying binary tree, and not on the topology of the tree itself. These results are analogous to classic results for the Cavender-Farris-Neyman model without a molecular clock. However, new techniques are required because the molecular clock assumption destroys the toric fiber product structure that governs group-based models without the molecular clock.

math.AG↗

arXiv2020

The $h^*$-polynomial of the order polytope of the zig-zag poset

We describe a family of shellings for the canonical triangulation of the order polytope of the zig-zag poset. This gives a new combinatorial interpretation for the coefficients in the numerator of the Ehrhart series of this order polytopein terms of the swap statistic on alternating permutations.

math.CO↗

arXiv2019

Identifiability in Phylogenetics using Algebraic Matroids

Identifiability is a crucial property for a statistical model since distributions in the model uniquely determine the parameters that produce them. In phylogenetics, the identifiability of the tree parameter is of particular interest since it means that phylogenetic models can be used to infer evolutionary histories from data. In this paper we introduce a new computational strategy for proving the identifiability of discrete parameters in algebraic statistical models that uses algebraic matroids naturally associated to the models. We then use this algorithm to prove that the tree parameters are generically identifiable for 2-tree CFN and K3P mixtures. We also show that the $k$-cycle phylogenetic network parameter is identifiable under the K2P and K3P models.

math.CO↗

arXiv2019

Exchangeable and Sampling Consistent Distributions on Rooted Binary Trees

We introduce a notion of finite sampling consistency for phylogenetic trees and show that the set of finitely sampling consistent and exchangeable distributions on n leaf phylogenetic trees is a polytope. We use this polytope to show that the set of all exchangeable and infinite sampling consistent distributions on 4 leaf phylogenetic trees is exactly Aldous' beta-splitting model and give a description of some of the vertices for the polytope of distributions on 5 leaves. We also introduce a new semialgebraic set of exchangeable and sampling consistent models we call the multinomial model and use it to characterize the set of exchangeable and sampling consistent distributions.

math.CO↗