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 ).

Density function

The probability density function of a rectified Gaussian distribution, for which random variables X having this distribution are displayed as , is given by

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

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

is the Dirac delta function

and, is the unit step function:

Alternative form

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

then,

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

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