Eliana Duarte

Research

I use algebra, geometry and combinatorics to understand statistical models: what they look like, how to represent them, and when their estimators have closed-form solutions.

Staged trees and context-specific models

Staged trees represent statistical models in which some independences hold only in certain contexts. I study their defining equations, when two trees describe the same model, and how to learn them from observational and interventional data.

With Liam Solus, Christiane Görgen, Lamprini Ananiadi
Key papers Representation of context-specific causal models · Equations defining probability tree models · Gröbner bases for staged trees

Nodes of the same colour share a stage.

Graphical and causal models

Graphical models encode dependence between random variables with a graph. I study their algebraic structure, from discrete models with interventions to Gaussian polytrees, and recently to quantum many-body systems.

With Liam Solus, Yulia Alexandr, Julian Vill, Daniele Tramontano, Mathias Drton, Dmitrii Pavlov, Maximilian Wiesmann
Key papers Algebraic geometry of discrete interventional models · Learning linear Gaussian polytree models with interventions · Algebraic geometry of quantum graphical models

Maximum likelihood and algebraic optimization

Which discrete statistical models have a maximum likelihood estimator given by a rational formula? This question connects statistics with toric geometry and polytopes. I also study related optimization problems, such as finding the closest point to a model in polyhedral norms.

With Orlando Marigliano, Bernd Sturmfels, Isobel Davies, Irem Portakal, Miruna-Ştefana Sorea, Nidhi Kaihnsa, Julia Lindberg, Angélica Torres, Madeleine Weinstein
Key papers Discrete statistical models with rational MLE · Polytopes with rational linear precision · Algebraic distance optimization in polyhedral norms

Geometric modeling and implicitization

Curves and surfaces in computer-aided design are given by parametrizations. Finding their implicit equations uses syzygies and free resolutions from commutative algebra.

With Hal Schenck, Alexandra Seceleanu, Benjamin Hollering, Maximilian Wiesmann
Key papers Tensor product surfaces and linear syzygies · Implicitization via virtual projective resolutions · Toric fiber products in geometric modeling

Software

staged-tree

A formalization of staged trees in the Lean 4 proof assistant.

Rational MLE

Code accompanying Discrete statistical models with rational MLE.