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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

kjetil b halvorsen
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Tomas Greif
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  • you can take a look at this introductory chapter for the derivation. https://www.stat.cmu.edu/~cshalizi/uADA/12/lectures/ch12.pdf – Bisaso Kuteesa Feb 25 '17 at 20:14
  • I removed the [tag:negative-binomial] since that is for negative-binomial distribution, not for negative binomial log likelihood ... an unfortunate clash of terminology – kjetil b halvorsen Nov 08 '18 at 14:49

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