The test error rate of the Bayes classifier on a set of data is known as the Bayes error rate. The Bayes error rate is the minimum possible test error rate for any classifier on that set of data.

Formula for Bayes error rate:

$1 - E(max_{j} Pr(Y = j|X))$

Derivation: Given an observation with predictor vector $x_o$, the Bayes classifier always assigns class $j$ to the observation where $j$ is the most likely class of the observation given $x_o$. Formally, it picks $j$ for which $Pr(Y = j | X = x_o)$ is maximal. It then follows that the error rate for predicting an observation with predictor vector $x_o$ is simply the probability that the observation belongs to a class other than the most likely given $x_o$. Formally, this probability is $1 - max_jPr(Y = j | X = x_o)$. Finally, this means that the Bayes classifier has an error rate of $1 - E(max_jPr(Y = j | X ))$, which is the average of the error rates for every possible of X.