"The noise parameter is built into the Bernoulli formulation."

Andy and colleagues wrote the following in a 2000 paper in Applied Statistics:

When there is this logistic regression parameterization in the Stan manual:

data {
  int<lower=0> N;
  vector[N] x;
  int<lower=0,upper=1> y[N];
parameters {
  real alpha;
  real beta;
model {
  y ~ bernoulli_logit(alpha + beta * x);

to what extent is it safe to assume that the model is equivalent to a latent parameterization where there is an implied residual epsilon{i} for each observation, where the residuals are distributed with a variance of π^2/3, as also described in the Austin and Merlo tutorial here. @Bob_Carpenter @andrewgelman

I would say that it is safe. I think it is easiest to understand if you look at the cdf of the logistic distribution (wikipedia link). When you set mu=0 and s=1, then the cdf becomes the inverse logit function and the variance of the logistic distribution is \pi^2/3. Then you have something like

\begin{align} P(y_i = 1 | B) &= P(X_i B + \epsilon_i > 0) \\ &= P(\epsilon_i > -X_i B) \\ & = 1 - 1/(1 + e^{-(-X_i B)}) \\ &= 1/(1 + e^{-X_iB}) \end{align}