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Trajectory inference methods are essential for analyzing the developmental paths of cells in single-cell sequencing datasets. It provides insights into cellular differentiation, transitions, and lineage hierarchies, helping unravel the dynamic processes underlying development and disease progression. However, many existing tools lack a coherent statistical model and reliable uncertainty quantification, limiting their utility and robustness. In this paper, we introduce VITAE (Variational Inference for Trajectory by AutoEncoder), a statistical approach that integrates a latent hierarchical mixture model with variational autoencoders to infer trajectories. The statistical hierarchical model enhances the interpretability of our framework, while the posterior approximations generated by our variational autoencoder ensure computational efficiency and provide uncertainty quantification of cell projections along trajectories. Specifically, VITAE enables simultaneous trajectory inference and data integration, improving the accuracy of learning a joint trajectory structure in the presence of biological and technical heterogeneity across datasets. We show that VITAE outperforms other state-of-the-art trajectory inference methods on both real and synthetic data under various trajectory topologies. Furthermore, we apply VITAE to jointly analyze three distinct single-cell RNA sequencing datasets of the mouse neocortex, unveiling comprehensive developmental lineages of projection neurons. VITAE effectively reduces batch effects within and across datasets and uncovers finer structures that might be overlooked in individual datasets. Additionally, we showcase VITAE’s efficacy in integrative analyses of multiomic datasets with continuous cell population structures. 

Structural causal models (SCMs) are widely used in various disciplines to represent causal relationships among variables in complex systems. Unfortunately, the true underlying directed acyclic graph (DAG) structure is often unknown, and determining it from observational or interventional data remains a challenging task. However, in many situations, the end goal is to identify changes (shifts) in causal mechanisms between related SCMs rather than recovering the entire underlying DAG structure. Examples include analyzing gene regulatory network structure changes between healthy and cancerous individuals or understanding variations in biological pathways under different cellular contexts. This paper focuses on identifying functional mechanism shifts in two or more related SCMs over the same set of variables -- without estimating the entire DAG structure of each SCM. Prior work under this setting assumed linear models with Gaussian noises; instead, in this work we assume that each SCM belongs to the more general class of nonlinear additive noise models (ANMs). A key contribution of this work is to show that the Jacobian of the score function for the mixture distribution allows for identification of shifts in general non-parametric functional mechanisms. Once the shifted variables are identified, we leverage recent work to estimate the structural differences, if any, for the shifted variables. Experiments on synthetic and real-world data are provided to showcase the applicability of this approach. 

Structural causal models (SCMs) are widely used in various disciplines to repre- sent causal relationships among variables in complex systems. Unfortunately, the underlying causal structure is often unknown, and estimating it from data remains a challenging task. In many situations, however, the end goal is to localize the changes (shifts) in the causal mechanisms between related datasets instead of learn- ing the full causal structure of the individual datasets. Some applications include root cause analysis, analyzing gene regulatory network structure changes between healthy and cancerous individuals, or explaining distribution shifts. This paper focuses on identifying the causal mechanism shifts in two or more related datasets over the same set of variables—without estimating the entire DAG structure of each SCM. Prior work under this setting assumed linear models with Gaussian noises; instead, in this work we assume that each SCM belongs to the more general class of nonlinear additive noise models (ANMs). A key technical contribution of this work is to show that the Jacobian of the score function for the mixture distribution allows for the identification of shifts under general non-parametric functional mechanisms. Once the shifted variables are identified, we leverage recent work to estimate the structural differences, if any, for the shifted variables. Experiments on synthetic and real-world data are provided to showcase the applicability of this approach. Code implementing the proposed method is open-source and publicly available at https://github.com/kevinsbello/iSCAN. 

Abstract The research on gradual typing has led to many variations on the Gradually Typed Lambda Calculus (GTLC) of Siek & Taha (2006) and its underlying cast calculus. For example, Wadler and Findler (2009) added blame tracking, Siek et al . (2009) investigated alternate cast evaluation strategies, and Herman et al . (2010) replaced casts with coercions for space efficiency. The meta-theory for the GTLC has also expanded beyond type safety to include blame safety (Tobin-Hochstadt & Felleisen, 2006), space consumption (Herman et al ., 2010), and the gradual guarantees (Siek et al ., 2015). These results have been proven for some variations of the GTLC but not others. Furthermore, researchers continue to develop variations on the GTLC, but establishing all of the meta-theory for new variations is time-consuming. This article identifies abstractions that capture similarities between many cast calculi in the form of two parameterized cast calculi, one for the purposes of language specification and the other to guide space-efficient implementations. The article then develops reusable meta-theory for these two calculi, proving type safety, blame safety, the gradual guarantees, and space consumption. Finally, the article instantiates this meta-theory for eight cast calculi including five from the literature and three new calculi. All of these definitions and theorems, including the two parameterized calculi, the reusable meta-theory, and the eight instantiations, are mechanized in Agda making extensive use of module parameters and dependent records to define the abstractions.