Differentiable Structural EM for Explicit-Latent Linear–Gaussian DAGs

We bridge two largely separate lines of work on directed graphical models: Classical Structural EM, which handles hidden variables and is commonly instantiated through search over discrete graph structures; and Continuous DAG learning methods, which avoid combinatorial search through smooth optimization, but are usually designed for fully observed data. We combine these ideas for linear–Gaussian models with explicit hidden variables. Our proposed method treats the graph weights as parameters of a generative Gaussian model over both observed and hidden variables. This makes the E-step exact: instead of filling in hidden variables with point estimates, it computes the needed posterior first and second moments. The M-step then uses smooth sparsity and acyclicity penalties to guide the graph search, but accepts an update only when it does not decrease the unpenalized expected complete-data objective. This preserves the usual monotone improvement of EM while still benefiting from continuous DAG optimization. Experiments show that both ingredients are necessary: exact second-moment inference in the E-step and the unpenalized acceptance check in the M-step. On hidden-confounding synthetic models and a linear–Gaussian version of Friedman’s hidden-variable benchmark, a spectral warm start helps the explicit-latent model recover hidden structure, while observed-only and random-initialized alternatives perform substantially worse.