Overview of the conditional independence tests implemented in bnlearn, with the respective reference publications.
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
\(2n\) factor) and is related to the deviance of the tested models.
The asymptotic \(\chi^2\) test (mi and
mi-adf, with adjusted degrees of freedom), the Monte Carlo
permutation test (mc-mi), the sequential Monte Carlo permutation
test (smc-mi), and the semiparametric test (sp-mi) are
implemented.
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.
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
\(2n\) factor). 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.
Available conditional independence tests (and the respective labels) for hybrid 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
\(2n\) factor). Only the asymptotic \(\chi^2\) test
(mi-cg) is implemented.