Causal discovery now reads direction from a symbolic-regression posterior: causalBMS tops 13 of 21 CauseMe dataset-metric pairs and returns closed-form mechanisms
Synopsis
Causal discovery no longer has to rest on conditional-independence tests or a single best-scoring model. causalBMS scores each candidate parent by how often it appears in expressions sampled from a Bayesian symbolic-regression posterior, so it still gives a direction in bivariate, strongly autocorrelated and high-noise settings. On the CauseMe benchmark it ranked first in 13 of 21 dataset-metric pairs and second in four more, while also returning closed-form functional forms.
Interpretation
Causal direction can now be read from a set of sampled closed-form expressions instead of first selecting one best model and reading edges off it. Established constraint-based and score-based methods typically recover only a Markov equivalence class, and in the bivariate case the marginal independence structure is the same in both directions, so those methods give no signal. On the CauseMe benchmark, causalBMS ranked first in 13 of 21 dataset-metric pairs and second in four more, staying close to the top elsewhere.
The method samples expressions with Bayesian symbolic regression and uses variable inclusion frequency as the causal score. The Bayesian machine scientist runs a Markov chain Monte Carlo sampler over closed-form expression trees, scores each expression by description length, and returns posterior samples rather than a single equation. In a bivariate nonlinear dynamical system combining self-excitation with unidirectional coupling, lagged scores for true causal links stay near 1 and near 0 for absent links across noise levels.
Beyond ranking edges, the method returns the closed-form mechanism that produced the data, together with a residual diagnostic that needs no ground truth. PCMCI+, VAR-LiNGAM, the linear Granger test and convergent cross mapping return only edge weights. On a four-variable biogeoscience structural causal model, the expression recovered for shortwave radiation matches the true mechanism, with error only in the coefficients.
Perspective
The demonstrated scope is synthetic: a bivariate nonlinear dynamical system with self-excitation and unidirectional coupling, the CauseMe benchmark suite, and simulations of a four-variable biogeoscience structural causal model under four noise regimes. The intended reader works on time-series causal inference or equation discovery.
The consistency result assumes the true mechanism is representable in closed form; outside that function class the inclusion frequencies remain a feature-selection signal but lose the consistency guarantee. The noise threshold depends on quantities involving the unknown true model and must be estimated from data, so it serves as a diagnostic rather than a hard guarantee. In the four-variable case, the recovered expressions for temperature and respiration contain extra terms with incorrect causal parents and are more complex than the true mechanisms.