Why non-Gaussianity alone runs out of information
Linear structural equation models describe each observed variable as a weighted sum of earlier variables plus an external disturbance. With Gaussian disturbances, observational data carry information mainly through the covariance structure, which generally cannot single out one directed graph; methods typically recover a Markov equivalence class and need extra assumptions such as the causal order. LiNGAM takes a different route: it treats disturbances as non-Gaussian and solves the linear acyclic model as a blind source separation problem, where the disturbances are independent sources and the mixing matrix encodes the causal coefficients.
That route has a hard ceiling. Classical LiNGAM models disturbances as random variables sampled independently across observations, so the only statistical diversity available to the separation algorithm is higher-order statistics, and identifiability requires that at most one disturbance be Gaussian. When several disturbances are Gaussian, higher-order statistics have nothing to exploit, and any temporal structure in the disturbances is simply discarded.
Mutual information rate brings time into identification
The paper extends the disturbance model from mutually independent random variables to mutually independent random processes, whose samples need not be independent over time, while keeping the same linear acyclic causal structure. Temporally independent disturbances remain a special case. Because the disturbances may now be sample-dependent, their temporal structure becomes an additional source of statistical diversity alongside higher-order statistics.
To use that diversity, the objective changes. Instead of minimizing the instantaneous mutual information among source estimates, the method minimizes the mutual information rate, defined through the entropy rate of each source estimate, which measures the per-sample density of the average uncertainty of a random process. The mutual information rate jointly incorporates higher-order statistics and sample dependence. For real, zero-mean wide-sense-stationary Gaussian disturbances, sources become distinguishable by second-order statistics when their autocovariance functions are nonproportional, meaning no nonzero scaling makes one autocovariance function equal another across all lags. This is the condition that lets Gaussian disturbance processes be identified, which the non-Gaussianity requirement alone forbids.
Two diversities fail in different regimes
The synthetic experiments were designed to separate the two information sources rather than to crown a single winner. Four settings were tested: HOS-only, HOS-dominant, temporal-dominant, and temporal-only. Performance was measured against sample size T from 0.5 to 10 thousand, using relative Frobenius error, where lower is better, and AUPRC, where higher is better. Curves report mean metric values over 50 trials, with shaded regions showing 95% confidence intervals.
The pattern is complementary rather than redundant. Blind source separation methods relying solely on higher-order statistics fail when the disturbance processes are Gaussian, because non-Gaussianity is insufficient for identification. Methods relying only on sample dependence perform poorly when temporal diversity is limited. Joint exploitation of both achieves near-perfect causal recovery across the tested disturbance settings. The practical reading is that the two diversities cover each other's blind spots: neither is a strictly better substitute for the other, and the choice of method should follow which diversity the data actually supply.
What the fMRI bootstrap stability score measures
On real functional magnetic resonance imaging data, the ground-truth causal structure is unknown, so the paper compares methods by the stability of their estimated causal relationships rather than by accuracy. Subject-level bootstrap resampling draws K subjects with replacement and concatenates their representations along the temporal axis to form a group representation. Stability is quantified across B = 500 bootstrap samples. For each bootstrap adjacency matrix, the transitive closure determines ancestral relationships among the selected independent components, and the probability that variable i is an ancestor of variable j is the rate at which that relation occurs across bootstraps.
The summary score S_anc is the mean absolute difference between opposing ancestral frequencies, computed over all variable pairs. Larger values indicate a more reproducible preference for one ancestral direction. Low values can reflect either directional ambiguity or frequent incomparability, whose frequency is tracked separately; incomparability was generally infrequent in these data, so the score primarily reflects directional consistency. This distinction matters because a high stability score is not the same as a correct graph: it measures reproducibility of direction, not agreement with an unknown truth.
Joint exploitation leads ancestral stability
ERBM, which exploits both higher-order statistics and sample dependence, yields the highest directional ancestral stability at S_anc = 0.7330, followed by WASOBI at 0.5721, DirectLiNGAM at 0.4504, and FastICA at 0.4170. ERBM consistently places the AU components downstream of SC, with all SC-to-AU ancestral probabilities exceeding 0.90. WASOBI favors the same pattern with less uniformity. Both methods also assign AU1 as an ancestor of AU2 in all estimates.
Greater bootstrap stability does not imply agreement across methods. ERBM favors SC3 as an ancestor of SC2 with probability 0.79, whereas WASOBI favors the opposite direction with probability 0.86. The paper's own reading is that temporal information improves directional reproducibility without necessarily establishing a unique causal graph. That is a narrower claim than recovering the true structure, and it is the claim the evidence supports.
Where the identifiability claim stops
The broader identifiability result is conditional. Gaussian disturbance processes become identifiable when their covariance functions are sufficiently distinct, formalized as nonproportional autocovariance functions; if the temporal structure does not separate the sources, sample dependence adds nothing. The synthetic evidence covers the tested disturbance settings and sample sizes, not every possible regime.
The fMRI analysis carries its own boundaries. Causal interpretation remains limited by the assumed common group-level structure, by subject-boundary artifacts introduced when concatenating subject representations along the temporal axis, and by unobserved confounders outside the selected SC and AU components. The stability score also depends on the bootstrap design, including the number of resamples and the choice of components. These constraints qualify the fMRI result specifically; they do not by themselves undermine the synthetic identifiability argument, which rests on the mutual information rate formulation.
The priority is matching diversity to the data
The decision this paper supports is not "always combine both" but "check which diversity your disturbances actually provide." If disturbances are Gaussian, higher-order statistics cannot identify the structure no matter how much data is collected, and the relevant question becomes whether their autocovariance functions are nonproportional. If temporal diversity is weak, sample dependence will not rescue identification, and non-Gaussianity remains the usable signal. The joint method is the right default when both are present, which is the regime where it produced near-perfect recovery in the tested settings.
For the fMRI application, the paper's own comparison sets the priority: among the four methods, the one jointly exploiting higher-order statistics and sample dependence gave the most reproducible ancestral directions, with S_anc = 0.7330 against 0.5721, 0.4504, and 0.4170. The most useful thing a reader can take from this study is that directional reproducibility and directional agreement are separate questions, and that the method choice should follow the disturbance regime rather than a general preference for one statistical tool.
Sources used in this answer
Sample Dependence-Aware Blind Source Separation for Linear Causal Discovery
Replacing the mutual information cost with a mutual information rate extends LiNGAM-style identifiability to Gaussian disturbance processes with nonproportional covariance functions, and jointly exploiting higher-order statistics and sample dependence yields near-perfect causal recovery in synthetic tests and higher bootstrap ancestral stability on fMRI data.