Functional Choice for p in a Bernoulli Distribution

Question

Why is the functional choice $p = \exp(x)/(1+\exp(x))$ to model $p$ a good one in a Bernoulli distribution?

Is it because it is limited at $0$ as $x$ approaches $0$ and $1$ as $x$ approaches infinity?

Answer 1

The logistic and logit transformations have advantages such as:

• (as you identified) symmetric in a sense between (0,1) and the whole real line
• analytically tractable: integral and derivatives have closed forms
• a natural log-odds interpretation: $x=\log\left(\frac{p}{1-p}\right)$
• the logistic distribution being close to a normal distribution (in practice logit and probit models produce very similar results apart from a scale factor on some of the parameters)