Problem understanding probabilistic generative models for classification
My understanding is:

The first line is OK - it derives from Bayes Rule

Assume that this probability follows a logistic function, that is that


$$P(C_1|x) = \frac{1}{1+exp(-a)}$$

Then if

$$ y = \frac{1}{1+exp(-z)}$$
then
$$ z = ln(\frac{y}{1-y})$$
(some lines below but with $a$ and $\sigma$)

Therefore:

$$ a = ln(\frac{P(C_1|x)}{1-P(C_1|x)})$$

If there are only two classes then $1-P(C_1|x)= P(C_2|x)$.

Then


$$a = ln(\frac{P(C_1|x)}{P(C_2|x)}$$

Using Bayes on $P(C_1|x)$ and $P(C_2|x)$

$$a = ln(\frac{P(x|C_1) P(C_1)}{P(x|C_2)P(C_2)}$$

The whole thing now is to model $P(x|C_i)$. He assumes that it is a normal distribution. And from there you should the second set of equations. The important part (as far as I remember) is that the two covariance matrix (for $C_1$ and for $C_2$) are the same. I do not have access to the book now but I will look it up tomorrow and see if there is any difficult step in the derivations.

tldr: The whole point is he assumes that the probability as a logistic, then he gets the formula for the $a$, then he assumes a normal distribution for $P(x|C_i)$, then he assumes same covariance matrix.