We discuss the role computational algebraic geometry and symbolic computation has played in regards to statistical problems related to inferring phylogenetic networks with a focus on identifiability. This article is an accompanying extended abstract of my talk at ISSAC 2024. The article summarizes work completed with several co-authors, including: Leo van Iersel, Remie Janssen, Mark Jones, Robert Krone, Colby Long, Samuel Martin, and Yukihiro Murakami.

Given a statistical model, the maximum likelihood degree is the number of complex solutions to the likelihood equations for generic data. We consider discrete algebraic statistical models and study the solutions to the likelihood equations when the data contain zeros and are no longer generic. Focusing on sampling and model zeros, we show that, in these cases, the solutions to the likelihood equations are contained in a previously studied variety, the likelihood correspondence. The number of these solutions give a lower bound on the ML degree, and the problem of finding critical points to the likelihood function can be partitioned into smaller and computationally easier problems involving sampling and model zeros. We use this technique to compute a lower bound on the ML degree for 2 x 2 x 2 x 2 tensors of border rank ≤ 2 and 3 x n tables of rank ≤ 2 for n = 11, 12, 13, 14, the first four values of n for which the ML degree was previously unknown.