# Rectified Gaussian distribution

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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 ${\displaystyle (0,\infty )}$).

## Density function

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

${\displaystyle f(x;\mu ,\sigma ^{2})=\Phi (-{\frac {\mu }{\sigma }})\delta (x)+{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\;e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}{\textrm {U}}(x).}$
A comparison of Gaussian distribution, rectified Gaussian distribution, and truncated Gaussian distribution.
${\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt\quad x\in \mathbb {R} ,}$
${\displaystyle \delta (x)={\begin{cases}+\infty ,&x=0\\0,&x\neq 0\end{cases}}}$
${\displaystyle {\textrm {U}}(x)={\begin{cases}0,&x\leq 0,\\1,&x>0.\end{cases}}}$

## Alternative form

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

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

then,

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

## 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 [1] proposed a variational learning algorithm for the rectified factor model, where the factors follow a mixture of rectified Gaussian; and later Meng [2] 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

1. Template:Cite doi
2. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
1. REDIRECT Template:Probability distributions