Youla–Kucera parametrization

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (constant 0) and a continuous distribution (a truncated Gaussian distribution with interval $(0,\infty )$).

Density function

The probability density function of a rectified Gaussian distribution, for which random variables X having this distribution are displayed as $X\sim {\mathcal {N}}^{\textrm {R}}(\mu ,\sigma ^{2})$, is given by

$f(x;\mu ,\sigma ^{2})=\Phi (-{\frac {\mu }{\sigma }})\delta (x)+{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\;e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}{\textrm {U}}(x).$

A comparison of Gaussian distribution, rectified Gaussian distribution, and truncated Gaussian distribution.

Here, $\Phi (x)$ is the cumulative distribution function (cdf) of the standard normal distribution:

$\Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt\quad x\in \mathbb {R} ,$

$\delta (x)$ is the Dirac delta function

$\delta (x)={\begin{cases}+\infty ,&x=0\\0,&x\neq 0\end{cases}}$

and, ${\textrm {U}}(x)$ is the unit step function:

${\textrm {U}}(x)={\begin{cases}0,&x\leq 0,\\1,&x>0.\end{cases}}$

Alternative form

Often, a simpler alternative form is to consider a case, where,

$s\sim {\mathcal {N}}(\mu ,\sigma ^{2}),x={\textrm {max}}(0,s),$

then,

$x\sim {\mathcal {N}}^{\textrm {R}}(\mu ,\sigma ^{2})$

Application

A rectified Gaussian distribution is semi-conjugate to the Gaussian likelihood, and it has been recently applied to factor analysis, or particularly, (non-negative) rectified factor analysis. Harva proposed a variational learning algorithm for the rectified factor model, where the factors follow a mixture of rectified Gaussian; and later Meng proposed an infinite rectified factor model coupled with its Gibbs sampling solution, where the factors follow a Dirichlet process mixture of rectified Gaussian distribution, and applied it in computational biology for reconstruction of gene regulatory network.

References

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