5. Conditional Gaussian BIC Score

5.1. Summary

The Conditional Gaussian BIC Score is a BIC-type score for conditional Gaussian (CG) models with mixed continuous and discrete variables. It evaluates a DAG or CG structure by combining CG log-likelihood with a penalty on the number of parameters.

5.2. When to use

  • Data are a mix of continuous and discrete variables.

  • You assume a CG model: given the discrete variables, continuous variables are multivariate normal with means and covariances that may depend on discrete configurations.

  • You are using score-based or hybrid algorithms that support CG models.

5.3. Model class

  • Conditional Gaussian Bayesian networks with discrete parents and linear- Gaussian continuous components.

5.4. Score form (conceptual)

As with other BIC scores:

BIC = 2 * logL − k * ln(N)

where logL is the CG log-likelihood and k is the number of free parameters (conditional means, covariances, and discrete probabilities).

5.5. Parameters

Parameter (camelCase)

Description

penaltyDiscount

Double ≥ 0.0. The penalty multiplier “c” in the modified BIC-type criterion (for example, a score of the form 2·log-likelihood − c·k·log(N), where k is the number of free parameters and N is the sample size). Larger values impose a stronger complexity penalty and yield sparser graphs; smaller values allow denser graphs. Default is 2.0.

structurePrior

Double ≥ 0.0. Structure prior coefficient controlling a binomial-style prior on the number of parents per node. When 0.0 (default), the score uses essentially a flat structure prior. Larger values encode a stronger preference for a particular expected parent count and can bias the search toward graphs with that typical in-degree.

discretize

Boolean. If true, continuous variables are discretized when a conditional Gaussian likelihood would require integrating over a continuous parent with a discrete child (a backup “discretize X” strategy). If false, the score uses the exact conditional Gaussian integration whenever possible. Default is true.

numCategoriesToDiscretize

Integer ≥ 2. Number of categories used when discretizing continuous variables in the backup discretization step. Default is 3. Larger values give a finer discretization but increase the size of the conditional tables and the number of parameters.

minSampleSizePerCell

Integer ≥ 2. Minimum required sample size per configuration (cell) in the conditional Gaussian model. If the per-cell sample size is too small, the exact CG calculations become unstable, and the score may fall back to the discretization strategy. Default is 4.

5.6. Strengths

  • Properly accounts for mixed data types without discretizing continuous variables.

  • Integrates naturally with CG independence tests and CG learning algorithms.

5.7. Limitations

  • Requires enough data per discrete configuration.

  • Assumes linear-Gaussian behavior for continuous components.

5.8. References

  • Andrews, B., Ramsey, J., & Cooper, G. F. (2018). Scoring Bayesian networks of mixed variables. International Journal of Data Science and Analytics, 6(1), 3–18.