The Bayes Classifier is a classifier that simply assigns an observation to the most likely class (the class with the highest probability), given some predictor values:
In a binary classification problem, the Bayes classifier corresponds to predicting TRUE if
where is a vector of predictors for a given observation.
One maybe not obvious wrinkle is that estimating the parameters of the Bayes Classifier doesn’t require Bayesian methods.
Naive Bayes
The Naive Bayes Classifier is a flavor of the Bayes Classifier that assumes all predictors are independent of one another.