Half-Cauchy Distribution#

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

The Half-Cauchy distribution is a continuous probability distribution that is derived from the Cauchy distribution but is restricted to only positive values. It is characterized by a single scale parameter (\(\beta\)), which determines the width of the distribution. Similar to the Cauchy distribution, the Half-Cauchy distribution has undefined mean and variance, making it an example of a “pathological” distribution with heavy tails.

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

Key properties and parameters#

Support

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

Mean

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Variance

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Parameters:

  • \(\beta\) : (float) Scale parameter, \(\beta > 0\).

Probability Density Function (PDF)#

\[ f(x|\beta) = \frac{2}{\pi \beta \left[1 + \left(\frac{x}{\beta}\right)^2\right]} \]
../../_images/0fdaac1beeaa0689ff23d51131e3fd519e1158a7312fe49238f4db156b077795.png

Cumulative Distribution Function (CDF):#

\[ F(x|\beta) = \frac{2}{\pi} \arctan\left(\frac{x}{\beta}\right) \]
../../_images/cd168752c2febd28a74514d752c511293e710234de40ed9d83b5cd78e87c244f.png

See also

Common Alternatives:

  • Half-Student’s t - The Half-Cauchy distribution is a special case of the Half-Student’s t-distribution with \(\nu=1\).

  • Half-Normal - A distribution that considers only the positive half of the normal distribution.

Related Distributions:

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

References#