Conditional independence tests

Description

Overview of the conditional independence tests implemented in bnlearn, with the respective reference publications.

Details

Unless otherwise noted, the reference publication for conditional independence tests is:

Edwards DI (2000). Introduction to Graphical Modelling. Springer, 2nd edition.

Additionally, for continuous permutation tests:

Legendre P (2000). "Comparison of Permutation Methods for the Partial Correlation and Partial Mantel Tests." Journal of Statistical Computation and Simulation, 67:37–73.

and for semiparametric discrete tests:

Tsamardinos I, Borboudakis G (2010). "Permutation Testing Improves Bayesian Network Learning." Machine Learning and Knowledge Discovery in Databases, 322–337.

Available conditional independence tests (and the respective labels) for discrete Bayesian networks (categorical variables) are:

• Mutual information: an information-theoretic distance measure. It's proportional to the log-likelihood ratio (they differ by a factor of 2n) and is related to the deviance of the tested models. The asymptotic \chi^2 test ("mi" and "mi-adf"), the Monte Carlo permutation test ("mc-mi"), the sequential Monte Carlo permutation test ("smc-mi"), and the semiparametric test ("sp-mi") are implemented. Compared to "mi", "mi-adf" adjusts the degrees of freedom for structural zeroes and automatically favours independence if there are fewer than 5 observations per parameter.

• Shrinkage estimator for the mutual information ("mi-sh"): an improved asymptotic \chi^2 test based on the James-Stein estimator for the mutual information.

  Hausser J, Strimmer K (2009). "Entropy Inference and the James-Stein Estimator, with Application to Nonlinear Gene Association Networks." Statistical Applications in Genetics and Molecular Biology, 10:1469–1484.

• Pearson's X^2: the classical Pearson's X^2 test for contingency tables. The asymptotic \chi^2 test ("x2" and "x2-adf", with adjusted degrees of freedom), the Monte Carlo permutation test ("mc-x2"), the sequential Monte Carlo permutation test ("smc-x2") and semiparametric test ("sp-x2") are implemented.

Available conditional independence tests (and the respective labels) for discrete Bayesian networks (ordered factors) are:

• Jonckheere-Terpstra: a trend test for ordinal variables. The asymptotic normal test ("jt"), the Monte Carlo permutation test ("mc-jt") and the sequential Monte Carlo permutation test ("smc-jt") are implemented.

Available conditional independence tests (and the respective labels) for Gaussian Bayesian networks (normal variables) are:

• Linear correlation: Pearson's linear correlation. The exact Student's t test ("cor"), the Monte Carlo permutation test ("mc-cor") and the sequential Monte Carlo permutation test ("smc-cor") are implemented.

  Hotelling H (1953). "New Light on the Correlation Coefficient and its Transforms." Journal of the Royal Statistical Society: Series B, 15(2):193–225.

• Fisher's Z: a transformation of the linear correlation with asymptotic normal distribution. The asymptotic normal test ("zf"), the Monte Carlo permutation test ("mc-zf") and the sequential Monte Carlo permutation test ("smc-zf") are implemented.

• Mutual information: an information-theoretic distance measure. Again, it is proportional to the log-likelihood ratio (they differ by a factor of 2n). The asymptotic \chi^2 test ("mi-g"), the Monte Carlo permutation test ("mc-mi-g") and the sequential Monte Carlo permutation test ("smc-mi-g") are implemented.

• Shrinkage estimator for the mutual information ("mi-g-sh"): an improved asymptotic \chi^2 test based on the James-Stein estimator for the mutual information.

  Ledoit O, Wolf M (2003). "Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection." Journal of Empirical Finance, 10:603–621.

No conditional independence tests are available for zero-inflated Bayesian networks.

Available conditional independence tests (and the respective labels) for conditional Gaussian Bayesian networks (mixed discrete and normal variables) are:

• Mutual information: an information-theoretic distance measure. Again, it is proportional to the log-likelihood ratio (they differ by a factor of 2n). Only the asymptotic \chi^2 test ("mi-cg") is implemented.