A common theme in causal inference is learning causal relationships between observed variables, also known as causal discovery. This is usually a daunting task, given the large number of candidate causal graphs and the combinatorial nature of the search space.

We first generalize existing algebraic characterizations of acyclicity to a class of matrix polynomials.

We study a general framework of distributional computational graphs: computational graphs whose inputs are probability distributions rather than point values. We analyze the discretization error that arises when these graphs are evaluated using finite approximations of continuous probability distributions. Such an approximation might be the result of representing a continuous real-valued distribution using a discrete representation or from constructing an empirical distribution from samples (or might be the output of another distributional computational graph). We establish non-asymptotic error bounds in terms of the Wasserstein-1 distance, without imposing structural assumptions on the computational graph.

One of the grand challenges of cell biology is inferring the gene regulatory network (GRN) which describes interactions between genes and their products that control gene expression and cellular function. We can treat this as a causal discovery problem but with two non-standard challenges: (1) regulatory networks are inherently cyclic so we should not model a GRN as a directed acyclic graph (DAG), and (2) observations have significant measurement noise so for typical sample sizes, there will always be a large equivalence class of graphs that are likely given the data, and we want methods that capture this uncertainty. Existing methods either focus on challenge (1), identifying cyclic structure from dynamics, or on challenge (2) learning complex Bayesian posteriors over directed acyclic graphs, but not both. In this paper we leverage the fact that it is possible to estimate the ``velocity'' of the expression of a gene with RNA velocity techniques to develop an approach that addresses both challenges. Because we have access to velocity information, we can treat the Bayesian structure learning problem as a problem of sparse identification of a dynamical system, capturing cyclic feedback loops through time. We leverage Generative Flow Networks (GFlowNets) to estimate the posterior distribution over the combinatorial space of possible sparse dependencies. Our results indicate that our method learns posteriors that better encapsulate the distributions of cyclic structures compared to counterpart state-of-the-art Bayesian structure learning approaches.

We prove that $\Omega(s\log p)$ samples suffice to learn a sparse Gaussian directed acyclic graph (DAG) from data, where $s$ is the maximum Markov blanket size. This improves upon recent results that require $\Omega(s^{4}\log p)$ samples in the equal variance case. To prove this, we analyze a popular score-based estimator that has been the subject of extensive empirical inquiry in recent years and is known to achieve state-of-the-art results. Furthermore, the approach we study does not require strong assumptions such as faithfulness that existing theory for score-based learning crucially relies on. The resulting estimator is based around a difficult nonconvex optimization problem, and its analysis may be of independent interest given recent interest in nonconvex optimization in machine learning. Our analysis overcomes the drawbacks of existing theoretical analyses, which either fail to guarantee structure consistency in high-dimensions (i.e.