I am trying to understand how penalized logistic regression works and I got stuck with negative binomial log-likelihood. I understand the the first two formulas and the penalization part in the likelihood formula, but I cannot figure out how can be the first part of the formula derived.
Probability: $$Pr(G=2|X=x)=\frac{e^{β_0+β^Tx}}{1+e^{β_0+β^Tx}}$$
Log-odds transformaton: $$log\frac{Pr(G=2|X=x)}{Pr(G=1|X=x)}=β_0+β^Tx,$$
Negative binomial log-likelihood: $$min_{(β0,β)∈R^{p+1}}−[\frac{1}{N}∑_{i=1}^{N}y_i⋅(β_0+x^T_iβ)−log(1+e^{(β_0+x^T_iβ)})]+λ[(1−α)||β||_2^2/2+α||β||1]$$
Source: https://web.stanford.edu/~hastie/glmnet/glmnet_alpha.html#log
