Log-odds has a linear relationship with the independent variables, which is why log-odds equals a linear equation.
What about log of probability? How is it related to the independent variables? Is there a way to check the relationship?
Topic probability logistic-regression
Category Data Science
In logistic regression, it isn’t the case that the log-odds are linearly related to the features. We posit that such a relationship exists and then find the coefficients giving the best fit. That assumed linear relationship between the log-odds and the features might be an awful assumption, and that is why models like neural networks can be useful.
If you want to propose a binomial model with $\log(\mathbb E[Y\vert X])=X\beta$, feel free to do so. In fact, R has no trouble fitting such a model.
set.seed(2022)
N <- 100
x <- runif(N, 1, 2)
y <- rbinom(N, 1, 0.5)
L <- glm(y ~ x, family = binomial(link = log))
summary(L)
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