Bayesian networks (BNs), represented by directed acyclic graphs (DAGs), provide a principled framework for modeling complex dependencies among random variables. As data dimensionality increases into the tens of thousands, fitting and marginalizing a full BN becomes computationally prohibitive--particularly when inference is only needed for a small subset of variables. Estimation-collapsibility addresses this challenge by ensuring that directly fitting a submodel, obtained by ignoring non-essential variables, still yields exact inference on target variables. However, current DAG-based criterion for checking estimation-collapsibility is computationally intensive, involving exhaustive vertex searches and iterative removals. Additionally, practical applications typically identify the underlying DAG only up to its Markov equivalence class, represented by a completed partially directed acyclic graph (CPDAG). To bridge this gap, we introduce sequential c-simplicial sets--a novel graphical characterization of estimation-collapsibility directly applicable to CPDAGs. We further propose DSCS, a computationally efficient algorithm for verifying estimation-collapsibility within CPDAG framework that scales effectively to high-dimensional BNs. Extensive numerical experiments demonstrate the practicality, scalability, and efficiency of our proposed approach.

Time series graphical models have recently received considerable attention for characterizing (conditional) dependence structures in multivariate time series. In many applications, the multivariate series exhibit variable-partitioned blockwise dependence, with distinct patterns within and across blocks. In this paper, we introduce a new class of time series Gaussian chain graph models that represent contemporaneous and lagged causal relations via directed edges across blocks, while capturing within-block conditional dependencies through undirected edges. In the frequency domain, this formulation induces a cross-frequency shared group sparse plus group low-rank decomposition of the inverse spectral density matrices, which we exploit to establish identifiability of the time series chain graph structure. Building on this, we then propose a three-stage learning procedure for estimating the undirected and directed edge sets, which involves optimizing a regularized Whittle likelihood with a group lasso penalty to encourage group sparsity and a novel tensor-unfolding nuclear norm penalty to enforce group low-rank structure. We investigate the asymptotic properties of the proposed method, ensuring its consistency for exact recovery of the chain graph structure. The superior empirical performance of the proposed method is demonstrated through both extensive simulation studies and an application to U.S. macroeconomic data that highlights key monetary policy transmission mechanisms.

A growing body of work has begun to study intervention design for efficient structure learning of causal directed acyclic graphs (DAGs).

Inthis paper,we propose anovelframework for computing Shapleyvalues that generalizes recent work that aims to circumvent the independence assumption.