# Trying to understand Bayes Classifier and Bayes error rate

For instance if we have an outcome with 3 classes A,B, and C. Assuming $$X$$~$$Uniform(0,1)$$ and $$Y$$ is as follows

If $$x > 0.6$$

$$P(Y = A) = 0.4$$

$$P(Y = B) = 0.3$$

$$P(Y = C) = 0.3$$

Alternatively, if $$x <= 0.6$$

$$P(Y = A) = 0$$

$$P(Y = B) = 0.2$$

$$P(Y = C) = 0.8$$

What is the Bayes Classifier and Bayes Error and how does that change with different distributions for x, say if $$X$$~$$Normal(0,1)$$

The Bayes classifier chooses the class with maximum posterior probability, i.e. $$P(Y=c|X=x)$$ and you have all you need; in other words, you have the posterior probabilities for each possible $$x$$. For example, if $$X=0.2$$, $$P(Y=C|X=0.2)$$ is the maximum, and the class is $$C$$.
The Bayes error you make is the error when you decide incorrectly, based on the Bayes decision rule. For example, $$P(X>0.6)=0.4$$ if $$X\sim U[0,1]$$, and if $$X>0.6$$, we choose class $$A$$ as our decision. There is $$0.3+0.3=0.6$$ probability of the correct class being not $$A$$. Merging the other case yields to following formula, just waiting for the values to be substituted:
$$P(\epsilon)=P(\epsilon|X>0.6)P(X>0.6)+P(\epsilon|X\leq 0.6)P(X\leq 0.6)$$