Half-Normal Distribution#

Univariate, Continuous, Non-Negative, Asymmetric, Light-tailed

The Half-Normal distribution is a continuous probability distribution that is derived from the Normal distribution but is restricted to only positive values. It is characterized by a single scale parameter (\(\sigma\)), which determines the width of the distribution.

In Bayesian statistics, the Half-Normal distribution is commonly used as a prior for scale parameters.

Key properties and parameters#

Support

\(x \in [0, \infty)\)

Mean

\(\dfrac{\sigma \sqrt{2}}{\sqrt{\pi}}\)

Variance

\(\sigma^2 \left(1 - \dfrac{2}{\pi}\right)\)

Parameters:

  • \(\sigma\) : (float) Standard deviation of the distribution, \(\sigma > 0\).

  • \(\tau\) : (float) Precision of the distribution, \(\tau > 0\).

Alternative parametrization

The Half-Normal distribution has 2 alternative parameterizations. It can be defined in terms of the standard deviation (\(\sigma\)) or in terms of the precision (\(\tau\)).

The link between the 2 alternatives is given by:

\[ \tau = \frac{1}{\sigma^2} \]

Probability Density Function (PDF)#

\[ f(x|\sigma) = \sqrt{\dfrac{2}{\pi\sigma^2}} \exp\left(-\dfrac{x^2}{2\sigma^2}\right) \]
../../_images/halfnormal_0_1.png
../../_images/halfnormal_1_0.png

Cumulative Distribution Function (CDF)#

\[ F(x|\sigma) = \text{erf}\left(\dfrac{x}{\sigma\sqrt{2}}\right) \]

where erf is the error function.

../../_images/halfnormal_2_0.png
../../_images/halfnormal_3_0.png

See also

Common Alternatives:

  • Half-Cauchy - A distribution with heavier tails that considers only the positive half of the Cauchy distribution.

Related Distributions:

  • Normal - The parent distribution from which the Half-Normal is derived.

  • Half-Student’s t - As \(\nu \to \infty\), the Half-Student’s t-distribution converges to the Half-Normal distribution.

  • Truncated Normal - A Half-Normal distribution can be considered a special case of the Truncated Normal distribution with mean \(0\) and lower bound \(0\).

References#