Bayesian Causal Discovery in Directed Cyclic Graphs with Closed-Form Lag Bayes Factors

In coarsely sampled longitudinal data, feedback loops can appear as cyclic contemporaneous structure, yet most causal-discovery methods assume acyclicity. {Bayesian} directed cyclic graph models allow cycles by using non-{Gaussian} errors with lagged outcomes and exogenous covariates to identify contemporaneous structure. However, consecutive lags are often highly correlated. Existing methods place a binary indicator on each candidate lag, creating exponentially many lag patterns per edge; evidence for a true edge can split across correlated configurations and fall below selection thresholds (posterior fragmentation). We replace these indicators with one categorical variable recording edge absence or active lag, so lags compete within one state space. Under the {Gaussian} scale-mixture of {Laplace} errors, each candidate coefficient integrates out exactly, yielding closed-form {Bayes} factors from cached scalars with no matrix operations for lag comparison. We prove edge-and-lag selection consistency and show that the at-most-one-active-lag assumption is least costly where fragmentation is most severe; the method is a finite-sample regulariser for correlated-lag regimes, not a uniformly better alternative. Cached sufficient statistics and incremental Cholesky updates make the sampler 55–67 times faster without changing the target posterior. In the largest single-lag synthetic benchmark, the categorical encoding raises the true-positive rate from 0.83 to 0.99 and lowers structural Hamming distance from 6.0 to 0.3. On the Health and Retirement Study panel (17,883 individuals, 13 waves), both formulations share a four-edge contemporaneous core, but the categorical encoding yields larger autoregressive coefficients and a sparser contemporaneous graph than the independent encoding.