I read that logistic regression formula i.e.
$ \log\frac{P(C_1|X)}{P(C_2|X)}=w^Tx+w_o$ 
but this equation is true if we have $P(X|C_1)$ and $P(X|C_2)$ sampled from two Gaussian with the same covariance matrices ($\sum_1=\sum_2$); furthermore, we can derive  the sigmoid transformation $P(C_1|X) = \frac{1}{1+ \exp{(-(w^Tx+w_o))}}$ but the linearity is given by assumption our assumption on $P(X|C_1)$ and $P(X|C_2)$; the question is:
I have a strong assumption on the data generation $P(X|C_i) (i.e. Gaussian with same covariance matrix and then same "shape")
but in logistic regression we throw away all these assumption and use this model without taking care the original data generation (I undestand this from https://books.google.it/books?id=NHODBAAAQBAJ&pg=PA291&lpg=PA291&dq=logistic+regression+why+sigmoid+same+covariances&source=bl&ots=2RLqMTEhXZ&sig=qz3vCEgr3HMDfNCuM34elnypaSE&hl=it&sa=X&ved=0ahUKEwilxaKn-Z7YAhXDVxQKHdxUCRA4ChDoAQhFMAM#v=onepage&q=logistic%20regression%20why%20sigmoid%20same%20covariances&f=false , page 291); isn't this a very strange thing? We are using something build on an assumption and then we throw away this assumption and use this model however? It seems to me a very bias model, because seems to be true just for a class of data (generated with Gaussians with same covariace) but not for all others dataset; where is my mistake?