Theory

A MeDIL causal model (MCM) [MGW20] is a bipartite DAG whose vertex set is partitioned into unobserved latent variables and observed measurement variables, with edges directed only from latents to measurements. Each measurement has out-degree zero—it is caused by latents but does not cause other variables—and each latent causes at least one measurement. This strong causal insufficiency assumption encodes that all dependences among measurements are due to shared latent causes. Unlike standard factor analysis, MCMs impose no linearity or distributional constraints; identifiability is obtained instead through causal minimality, as described below.

Under strong causal insufficiency, the only conditional independence (CI) constraints among measurement variables are marginal independence constraints (Proposition 6 of [MGW20]). Structure learning therefore reduces to estimating the undirected dependence graph (UDG) over measurements—an edge connects \(M_i\) and \(M_j\) when they are marginally dependent—via pairwise nonparametric independence tests. Because many distinct MCMs may be observationally consistent with the same UDG, identifiability is obtained by applying causal minimality: the minimum MCM (minMCM) is the observationally consistent model with the fewest latent variables [MLAS23].

The central result (Proposition 7 of [MGW20]) is that every minMCM corresponds to a minimum edge clique cover (ECC) [GGfN09] of the UDG—a smallest collection of cliques whose union covers every edge—with one latent per clique directed to exactly the clique’s measurements. Two structural properties follow: the latents of any minMCM are pairwise marginally independent (Proposition 9), and a minMCM may require more latents than measurements (Proposition 10), unlike in standard factor analysis. Finding a minimum ECC is NP-hard; MeDIL provides both an exact branch-and-bound solver [GGfN09] and a polynomial-time heuristic based on the one-pure-child assumption [MLAS23].

The estimated biadj can be parameterized in the linear Gaussian setting via GaussianMCM, which fits a model in which each measurement is a weighted sum of its latent parents plus independent Gaussian noise, with weights and variances estimated by maximum likelihood. NeuroCausalFactorAnalysis (NCFA; [MLAS23]) instead learns a variational autoencoder (VAE) whose decoder is constrained to the structure of biadj, so each measurement’s generative network receives inputs only from its assigned latent causes, and parameters are estimated by maximizing the evidence lower bound (ELBO).