# Logistic regression link function from optimization perspective

I learned the logistic regression from optimization context instead of statistics context.

Logistic regression is trying to minimize logistic loss. I know there are two forms, but they are identical. Essentially from probability perspective, we are doing max likelihood on Bernoulli probability models.

Why there are two different logistic loss formulation / notations?

Difference between logit and probit models

Are we essentially changing logistic loss to something else? Why I never heard of "probit loss"?

• I tried to give an answer, as I come from a similar perspective! Note that you have a typo in the question title: missing "i" in link – GeoMatt22 Sep 19 '16 at 20:19
• @GeoMatt22 thanks for your answer and suggestions. typo fixed! – hxd1011 Sep 19 '16 at 20:20
• What is the question here exactly? Are you asking if there is a way to formulate bernoulli regression with a probit link as a loss of the form $\log(1+ \exp(−y \hat{y}))$, as per the question you linked? – Andrew M Sep 19 '16 at 20:48
• @AndrewM I was trying to ask what's the objective function for other link functions. and if there are any intuitive explanations for other objective functions. – hxd1011 Sep 20 '16 at 1:17

As I understand it*, link functions are associated with Generalized linear modeling (GLM). The link function is used to relate the (conditional) expected value of the dependent variable $y$ to a linear predictor constructed from the independent variables $x$, i.e. $$g[\langle y\mid x\rangle]=L[x,\theta]$$ where $g[\,]$ is the link function and $L[\,]$ is the linear predictor, parameterized by $\theta$, which is to be estimated by MLE, assuming i.i.d. data $y$. The link function is relatively unconstrained, but to be admissable it must have an appropriate domain and range, and it must be invertible.
For the case of Bernoulli distributed data $y\sim\mathrm{Bern}[p]$, the left hand side becomes $g[\,p[x]\,]$, where $p[x]=\langle y\mid x\rangle$ is the conditional mean.