Derivation of Bayes classifier equation

by Francesco Boi   Last Updated September 11, 2019 09:19 AM

In the Elements of Statistical learning book when introducing Linear Discriminant Analysis it says:

A simple application of Bayes theorem gives us

$Pr(G=K|X=x) = \frac{f_k(x)\pi_k}{\sum_{l=1}^Kf_l(x)\pi_l}$

where $\pi_k$ is the prior probability of class $k$ and $f_k(x)$ is the class conditional probability.

  1. What is the class conditional probability? Is it $Pr(X=x|G=K)$?
  2. How is derived the above equation from the Bayes theorem? I know $Pr(G=K|X=x) = \frac{Pr(X=x|G=K)Pr(G=K)}{Pr(X=x)}$

I know that $Pr(G=K)=\pi_k$ but I do not know how to derive the rest of the equation.

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