Under zero-one loss, a Bayes classifier assigns an observation to a class of greatest posterior probability. For two classes with prior probabilities and densities , it chooses class one exactly when .
For two multivariate normal densities, the Bayes log posterior odds are a quadratic polynomial. Equal covariance matrices cancel the quadratic term and give linear discriminant analysis; unequal covariance matrices give quadratic discriminant analysis.
A binary Bayes classifier under zero-one loss is unique up to null sets when the posterior class probabilities tie only on a null set. For two distinct nonsingular Gaussian distributions, the tie set is the zero set of a nonzero quadratic polynomial and has Lebesgue measure zero.
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A Bayes classifier is a statistical classification technique based on Bayes' theorem, which provides a way to update the probability of a hypothesis as more evidence or information becomes available. In the context of classification, the Bayes classifier classes a given sample into one of several classes based on the probabilities of the sample belonging to each class.