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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?

enter image description here

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  • $\begingroup$ I think you are asking about the "squishing" behavior of the sigmoid function for extreme values, those beyond, say, -5 and 5. If so, what exactly do you want to know about these extreme values? By the way, this paper may offer the insight you are looking for. $\endgroup$
    – mhdadk
    Nov 7, 2021 at 21:48
  • $\begingroup$ You appear to be trying to do something that would be quite simple with a statistical regression model. And "binary classifier outcome" doesn't mean very much. Remember that classifiers are forced choice dichotomizers which is seldom what is really needed to solve a problem. See for example this. $\endgroup$ Nov 7, 2021 at 22:54

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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}

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  • $\begingroup$ Thanks for the answer, I added clarification to my question. $\endgroup$
    – jnemecz
    Nov 7, 2021 at 21:37

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