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Bayes Optimal Classification Rule

Cross Validated Asked by John on November 28, 2020

Suppose that $Y∈ {0,1}$ and $P(Y=1) = 1/2$. The distribution of $X|Y= 0$ is discrete and is specified by $$P(X= 1|Y= 0) = 1/3 $$ $$P(X= 2|Y= 0) = 2/3$$The distribution of $X|Y= 1$ is discrete and is given by $$P(X= 2|Y= 1) = 1/3$$ $$P(X= 3|Y= 1) = 2/3$$

This is all I can find about the Bayes Classifier in my textbook:

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How would I be able to find the Bayes Classifier?

One Answer

Bayes classifier chooses the class with maximum posterior probability. Since there are two classes, if $P(Y=1|X=x)geq1/2$ it means posterior for class $1$ is the maximum one, and we choose it. That is why if $eta(X)geq 1/2$, it chooses class $1$.

The formula for the posterior you've shared at the end is correct. You just need to be careful about calling the negative class as $0$ or $-1$, and make sure $X$ values given in class conditional probabilities are correct (e.g. the upper one deals with $X=1,2$ and the lower one deals with $X=2,3$). In the end, you'll have a function that for each $X$ value, a class label is assigned.

Answered by gunes on November 28, 2020

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