Eliana Duarte

Eliana Duarte, Hal Schenck

Let U be a basepoint free four-dimensional subpace of the space of sections of bidegree (a,b) on X = P^1 x P^1, with a and b at least 2. The sections corresponding to U determine a regular map from X to P^3. We show that there can be at most one linear syzygy on the associated bigraded ideal I_U in the bigraded ring k[s,t;u,v]. Existence of a linear syzygy, coupled with the assumption that U is basepoint free, implies the existence of an additional "special pair" of minimal first syzygies. Using results of Botbol, we show that these three syzygies are sufficient to determine the implicit equation of the image of X in P^3; we also show that the singular locus must contain a line.

D. Tramontano, L. Waldmann, M. Drton et al.

We present a consistent and highly scalable local approach to learn the causal structure of a linear Gaussian polytree using data from interventional experiments with known intervention targets. Our methods first learn the skeleton of the polytree and then orient its edges. The output is a CPDAG representing the interventional equivalence class of the polytree of the true underlying distribution. The skeleton and orientation recovery procedures we use rely on second order statistics and low-dimensional marginal distributions. We assess the performance of our methods under different scenarios in synthetic data sets and apply our algorithm to learn a polytree in a gene expression interventional data set. Our simulation studies demonstrate that our approach is fast, has good accuracy in terms of structural Hamming distance, and handles problems with thousands of nodes.

Eliana Duarte, Liam Solus

We address the problem of representing context-specific causal models based on both observational and experimental data collected under general (e.g. hard or soft) interventions by introducing a new family of context-specific conditional independence models called CStrees. This family is defined via a novel factorization criterion that allows for a generalization of the factorization property defining general interventional DAG models. We derive a graphical characterization of model equivalence for observational CStrees that extends the Verma and Pearl criterion for DAGs. This characterization is then extended to CStree models under general, context-specific interventions. To obtain these results, we formalize a notion of context-specific intervention that can be incorporated into concise graphical representations of CStree models. We relate CStrees to other context-specific models, showing that the families of DAGs, CStrees, labeled DAGs and staged trees form a strict chain of inclusions. We end with an application of interventional CStree models to a real data set, revealing the context-specific nature of the data dependence structure and the soft, interventional perturbations.