\name{independence-tests}
\alias{independence-tests}
\alias{independence tests}
\title{Conditional independence tests}
\description{

  Overview of the conditional independence tests implemented in \pkg{bnlearn},
  with the respective reference publications.

}
\details{

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

  Edwards DI (2000). \emph{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." \emph{Journal of Statistical
    Computation and Simulation}, \strong{67}:37--73.

  and for semiparametric discrete tests:

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

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

  \itemize{

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

    \item \emph{Shrinkage estimator} for the \emph{mutual information}
      (\code{"mi-sh"}): an improved asymptotic \eqn{\chi^2}{chi-square} 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."
        \emph{Statistical Applications in Genetics and Molecular Biology},
        \strong{10}:1469--1484.

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

  }

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

  \itemize{

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

  }

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

  \itemize{

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

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

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

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

    \item \emph{Shrinkage estimator} for the \emph{mutual information}
      (\code{"mi-g-sh"}): an improved asymptotic \eqn{\chi^2}{chi-square} 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."
        \emph{Journal of Empirical Finance}, \strong{10}:603--621.

  }

  No conditional independence tests are available for \emph{zero-inflated
  Bayesian networks}.

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

  \itemize{

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

  }

}
\keyword{package}
