How do you achieve the sigmoid function step by step? I’ve read it’s the opposite of the logit function, so logit could be a starting point. Even to I don’t understand why we do the log to the odds formula either.
1 How do we achieve:
log(p/(1−p)) Inverse-> 1/(1+e^(-x))
2 And:
Why do we do the log of p/(1−p) And how can i intuitively see the why.
On
Set $x=\log\frac{p}{1-p}$. Then \begin{align*} x&=-\log \frac{1-p}{p}\\ -x&=\log\left(\frac{1}{p}-1\right)\\ e^{-x}&=\frac{1}{p}-1\\ e^{-x}+1&=\frac{1}{p}\\ p&=\frac{1}{1+e^{-x}}. \end{align*} The Wikipedia article on the logit function provides a nice history of the function and applications where it is used.
Somebody also told me this solution which I find easier