Math behind likelihood function for logit/probit models?

Question

Given a model $\hat{y_i} = F(x_i \hat{\beta}) + \epsilon_i$,

where F is some mapping function that could be, for example, the sigmoid function, and x is a row vector of your features

I see that Likelihood $\ell = \prod_{i | y_i=0}^N F(-(x_i\hat{\beta})) * \prod_{i | y_i=1}^N 1-F(-(x_i\hat{\beta}))$

But why would the inputs inside the F be negative? And also, if we are changing the i's for each product, why is it necessary to do $ 1 - F(-(x_i\hat{\beta})) $ instead of just $ F(-(x_i\hat{\beta})) $?

Answer 1

The negative sign is by symmetry of the logit and probit CDFs about zero:

$$ \begin{cases} F(-x) &= 1 - F(x)\\ 1 - F(-x) &= 1 - [1 - F(x)] = F(x) \end{cases} $$

Basically, the argument to $F$ doesn't have to be negative, and you could use the symmetry above to write:

$$ \mathcal{L} = \prod_n \left[ (1 - F(x_n \beta))^{1 - y_n} \times F(x_n \beta)^{y_n} \right] $$

We need to use $1-F$ instead of $F$ because the model says so. It assumes that the distribution of the binary response variable $Y_n \in \{0,1\}$ looks like this:

$$ \begin{cases} P(Y_n = 1 \mid X_n) &= F(X_n \beta)\\ P(Y_n = 0 \mid X_n) &= 1 - P(Y_n = 1 \mid X_n) =1 - F(X_n \beta) \end{cases} $$