How can be normalized value from logistic function used to determine probability of binary classifier outcome?

Question

I would like to discuss chapter that comes from Foreign-Exchange-Rate Forecasting With Artificial Neural Networks.

This chapter (see screenshot) describes a binary classifier made from neural network output values (this case predicts binary output). I understand meaning of these formulas but what I do not understand is equation 12.4 in the sense how can be normalized value $g_i^+(x)$ used as a probability value for positive outcome and $1 - g_i^-(x)$ for negative outcome prediction?

Q: How is strong connection between predicted value from neural network and probability of predicted binary outcome?

Edit1: @mhdadk: I am aware of this work with probability in binary output, my question was more about how relevant is the normalization with the sigmoid logical function itself to produce confidence / probability measure (probability density).

I'm thinking that, for example, if the input to normalization function is in the interval approx. $(-\infty, -5)$ or $(5,+\infty)$ will basically still be normalized to the value of 1 (simply and figuratively speaking). With respect to sigmoid function everything except values from the interval $(-5,5)$ will be "truncated" to the value very near of 1. And outside the interval $(-5,5)$ there is infinitely more values than inside this interval.

How does the probability, after conversion to the normalized value $(0,1)$, is significant in this manner?

Answer 1

Given two classes $c_1$ and $c_2$, if $$ p(c_1 \mid x) = g_i^+(x) $$ and since $$ p(c_1 \mid x) + p(c_2 \mid x) = 1 $$ then \begin{align} p(c_1 \mid x) + p(c_2 \mid x) &= 1 \\ p(c_2 \mid x) &= 1 - p(c_1 \mid x) \\ p(c_2 \mid x) &= 1 - g_i^+(x) \end{align}