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I know the equation of the sigmoid function and use it in logistic regression, SVM, etc.
$$ S(x) = \frac{1}{1 + e^{-x}} $$
In the case of the sigmoid function, What is the exact input and output of this function? What I know is, it takes the value of x which is written in the above equation. The value of x is the prediction and gives output in a probability value. The probability of a point belongs to a specific class.
Could anyone give me a better explanation of sigmoid function?
Your description is correct. The proper name of the function is logistic function, as "sigmoid" is ambiguous and may be applied to different S-shaped functions. It takes as input some value $x$ on real line $x \in (-\infty, \infty)$ and transforms it to the value in the unit interval $S(x) \in (0, 1)$. It is commonly used to transform the outputs of the models (logistic regression, neural networks) to probabilities, because probabilities are also bounded to unit interval. It is not the only function that does this, there are many others like probit, or cloglog, but logistic function is the most popular one.
$ is used to enclose inline formula, $$ to enclose the block one.
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If you have a probability $p$ with $0<p<1$ then the odds are $\frac{p}{1-p}$ and the log-odds are $\log\left(\frac{p}{1-p}\right)$.
Your $S(x)$ is the inverse function to this, so $S\left(\log\left(\frac{p}{1-p}\right)\right)=p$ and $\log\left(\frac{S(x)}{1-S(x)}\right)=x$.
It turns log-odds into probabilities.
