Rahul Sharma (Editor)

Rectified Gaussian distribution

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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 , ∞ ) ).

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Density function

The probability density function of a rectified Gaussian distribution, for which random variables X having this distribution are displayed as X ∼ N R ( μ , σ 2 ) , is given by

f ( x ; μ , σ 2 ) = Φ ( − μ σ ) δ ( x ) + 1 2 π σ 2 e − ( x − μ ) 2 2 σ 2 U ( x ) .

Here, Φ ( x ) is the cumulative distribution function (cdf) of the standard normal distribution:

Φ ( x ) = 1 2 π ∫ − ∞ x e − t 2 / 2 d t x ∈ R ,

δ ( x ) is the Dirac delta function

δ ( x ) = { + ∞ , x = 0 0 , x ≠ 0

and, U ( x ) is the unit step function:

U ( x ) = { 0 , x ≤ 0 , 1 , x > 0.

Alternative form

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

s ∼ N ( μ , σ 2 ) , x = max ( 0 , s ) ,

then,

x ∼ N R ( μ , σ 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

Rectified Gaussian distribution Wikipedia
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