logistic sigmoid:
\begin{equation}
\sigma(x) = \frac{1}{1+\exp(-x)}
\end{equation}
logistic sigmoid经常用于产生Bernoulli distribution的参数p，因为值域(0,1)
softplus function
\begin{equation}
\varsigma(x) = \log {(1+\exp(x))}
\end{equation}
softplus function常用语产生正态分布的$\beta$ 或$\sigma$参数，因为值域$(0,\infty)$
Useful Properties of logistic sigmoid and softplus function
\begin{equation}
\sigma(x) = \frac{\exp(x)}{\exp(x)+\exp(0)}
\end{equation}
\begin{equation}
\frac{d}{dx}\sigma(x) = \sigma(x)(1-\sigma(x))
\end{equation}
\begin{equation}
1-\sigma(x) = \sigma(-x)
\end{equation}
\begin{equation}
\log \sigma(x) =- \varsigma(-x)
\end{equation}
\begin{equation}
\frac{d}{dx}\varsigma(x) = \sigma(x)
\end{equation}
\begin{equation}
\forall x \in (0,1),\sigma^{-1}(x) = \log {(\frac{x}{1-x})}
\end{equation}
$\sigma^{-1}(x)$ is called the logit in statistics
\begin{equation}
\forall x > 0,\varsigma^{-1}(x) = \log {(\exp(x)-1)}
\end{equation}
\begin{equation}
\varsigma(x) = \int_{-\infty}^x \sigma(y)dy
\end{equation}
\begin{equation}
\varsigma(x) - \varsigma(-x)= x
\end{equation}