Algebraic Statistics in OSCAR
Tobias Boege, Antony Della Vecchia, Marina Garrote-López, Benjamin Hollering
TL;DR
The paper presents OSCAR's AlgebraicStatistics module, which treats parametric statistical models as images under polynomial maps and analyzes their Zariski closures $V=\overline{\mathcal{M}}$, enabling algebraic-statistical methods within a extensible Julia framework. It develops a unified, extensible interface for graphical models and phylogenetics, including Gaussian, discrete, and group-based models, as well as phylogenetic networks, with specialized structures such as color annotations and Fourier coordinates to simplify computations. A core contribution is the suite of implicitization techniques, including toric-structure exploits via 4ti2 and modern methods like multigraded and birational implicitization, delivering substantial speed-ups for vanishing-ideal computations. Additionally, the work integrates serialization and database functionality (mrdi/JSON, OscarDB) to share and reuse results, illustrating practical impact through benchmarks and ready-to-share data for small phylogenetic trees.
Abstract
We introduce the AlgebraicStatistics section of the OSCAR computer algebra system. We give an overview of its extensible design and highlight its features including serialization of data types for sharing results and creating databases, and state-of-the-art implicitization algorithms.