  In this study, both Bayesian classifiers and mutual information classifiers
are examined for binary classifications with or without a reject option. The
general decision rules in terms of distinctions on error types and reject types
are derived for Bayesian classifiers. A formal analysis is conducted to reveal
the parameter redundancy of cost terms when abstaining classifications are
enforced. The redundancy implies an intrinsic problem of "non-consistency" for
interpreting cost terms. If no data is given to the cost terms, we demonstrate
the weakness of Bayesian classifiers in class-imbalanced classifications. On
the contrary, mutual-information classifiers are able to provide an objective
solution from the given data, which shows a reasonable balance among error
types and reject types. Numerical examples of using two types of classifiers
are given for confirming the theoretical differences, including the
extremely-class-imbalanced cases. Finally, we briefly summarize the Bayesian
classifiers and mutual-information classifiers in terms of their application
advantages, respectively.
