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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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If $K$ is the number of classes, we can correct the formula as follows and use lowercase $k$ for referring to a specific class: $$P(G=k|X=x)=\frac{f_k(x)\pi(x)}{\sum_{l=1}^K f_l(x)\pi_l}$$

The class conditional probability is $P(X=x|G=k)$, or in case of non-discrete RVs class conditional density we use $f(X=x|G=k)$. The derivation follows from your note in (2): $$P(G=k|X=x)=\frac{\overbrace{P(X=x|G=k)}^{f_k(x)}\overbrace{P(G=k)}^{\pi_k}}{P(X=x)}$$ And, we can expand $P(X=x)$ via Total Probability Theorem: $$P(X=x)=\sum_{l=1}^K P(X=x|G=l)P(G=l)=\sum_{l=1}^Kf_l(x)\pi_l$$

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  • $\begingroup$ RV stands for real-valued? $\endgroup$ – Francesco Boi Sep 11 at 12:21
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    $\begingroup$ Random Variable $\endgroup$ – gunes Sep 11 at 12:30

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