Algebraic Statistics for Ecological and Biological Systems

Identifiability, estimability, and parameter reduction methods provide tools to understand the interactions between parameters, model structure, and outputs—and how these interactions determine what inferences and predictions are possible for a given system. In particular, issues of identifiability and uncertainty can affect whether it is possible to select an optimal intervention—an important question for applied infectious disease modeling.  In this talk, we will explore how identifiability can be used in practice to help inform epidemiological decision-making, and when intervention strategies are or are not robust to uncertainty in the model parameters and structure.

Factor analysis (FA) is a statistical tool for studying how observed variables with some mutual dependences can be expressed as functions of mutually independent unobserved factors, and it is widely applied throughout the psychological, biological, and physical sciences.

We revisit this classic method from the comparatively new perspective given by advancements in causal discovery and deep learning, introducing a framework for Neuro-Causal Factor Analysis (NCFA).

Our approach is fully nonparametric: it identifies factors via latent causal discovery methods and then uses a variational autoencoder (VAE) that is constrained to abide by the Markov factorization of the distribution with respect to the learned graph.
We evaluate NCFA on real and synthetic data sets, finding that it performs comparably to standard VAEs on data reconstruction tasks but with the advantages of sparser architecture, lower model complexity, and causal interpretability.

Unlike traditional FA methods, our proposed NCFA method allows learning and reasoning about the latent factors underlying observed data from a justifiably causal perspective, even when the relations between factors and measurements are highly nonlinear.
This talk will especially focus on the causal discovery process (including a Gröbner basis inspired search algorithm) and structural identifiability, and I will provide a basic introduction to the main concepts (causal discovery, VAEs, Gröbner bases, etc.) rather than assuming a specific background in either (causal) machine learning or algebraic statistics.

We will discuss a novel method to reconstruct phylogenetic networks based on algebraic invariants. While there is a long tradition of using algebraic invariants in phylogenetics, our work is the first to define phylogenetic invariants on concordance factors (frequencies of 4-taxon splits in the input gene trees) to identify level-1 phylogenetic networks under the multispecies coalescent model. Our novel inference methodology is optimization-free as it only requires the evaluation of polynomial equations, and as such, it bypasses the traversal of network space, yielding a computational speed at least 10 times faster than the fastest-to-date network methods. We illustrate the accuracy and speed of our new method on a variety of simulated scenarios as well as in the estimation of a phylogenetic network for the genus Canis.

Evolutionary convergence among lineages toward similar traits in similar circumstances is powerful evidence for biological adaptation, the exquisite fit of organisms to their environment caused by natural selection. Comparative methods allow biologists to test for convergence by comparing traits and other factors among extant lineages. Phylogenetic comparative methods (PCMs) arose in the 1980s out of recognition that nonindependence among lineages due to shared ancestry could result in statistical overconfidence in the evidence for convergence. One way to account for nonindependence adopted by many PCMs is to use phylogenies to predict the expected covariance among lineages due to shared ancestry. For continuous traits, the expected covariance is derived from stochastic diffusion models that embed assumptions about the evolutionary process. PCMs have progressed from using simpler to more complex models, aided by large increases in phylogenetic data and computational resources. More complex models not only include more biologically realistic assumptions, but expand the realm of questions that PCMs address. Today PCMs not only test for convergence, but instead address a wide variety of hypotheses about the tempo and mode of evolution. Larger data sets, increased sophistication of models, and better parameter estimation methods have contributed to widespread adoption of PCMs by empiricists. Despite this progress, there are unresolved fundamental issues that probably deserve more theoretical treatment, such as identifiability, parameter interpretation, model adequacy, models of complex traits, and model comparison. Future work will hopefully expand what we can learn using PCMs and identify epistemic limits to what can learn about the past when sampling is limited to extant lineages.

Gaussian graphical models are multivariate Gaussian statistical models in which a graph encodes conditional independence relations among the random variables. Adding colors to this graph allows us to describe situations where some entries in the concentration matrices in the model are assumed to be equal. In this talk, we focus on RCOP models, in which this coloring is obtained from the orbits of a subgroup of the automorphism group of the underlying graph. We show that when the underlying block graph is a one-clique-sum of complete graphs, the Zariski closure of the set of concentration matrices of an RCOP model on this graph is a toric variety. We also give a Markov basis for the vanishing ideal of this variety in these cases. Finally, we present preliminary results on the ideals of RCOP graphs which are not block graphs.

Brownian motion tree models (BMTM) are linear Gaussian covariance models that describe continuous trait evolutions along a phylogenetic tree. The maximum likelihood degree (ML degree) and the reciprocal maximum likelihood degree (RML degree) of a BMTM measure the algebraic complexity of solving the direct and reciprocal maximum likelihood estimation problem, respectively, for these models. We give explicit formulas for the RML degree of any BMTM and present preliminary results on the ML degree of a BMTM. This  includes the invariance of ML degree under the choice of the root in a BMTM and the ML degree of BMTMs with star tree structure. The talk is based on previous work with Boege, Coons, Eur and Rottgers and ongoing work with Coons, Cox and Nometa. 

The inference of phylogenetic networks, essential for understanding evolutionary relationships involving hybridization and horizontal gene transfer, presents formidable challenges in both theory and practice. While standard phylogenetic methods can infer gene trees from genetic data, these trees only indirectly reflect the species network topology due to horizontal inheritance and incomplete lineage sorting. 

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 will aim to fill these gaps and address both the identifiability of the full topology of the network and the numerical parameters by investigating the ideals defined by quartet concordance factors for topological semi-directed networks.