Half-Cauchy distribution

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The half-Cauchy is defined on and is Cauchy distributed on this domain. It can also be interpreted as a mixture of inverse gamma distributions[]]. In other words:

\begin{equation*}
  X \sim \operatorname{C}^{+}(0, a)
\end{equation*}

is equivalent to

\begin{align*}
  X^{2} &\sim  \operatorname{InverseGamma}(1/2,\, 1/\xi)\\
  \xi &\sim \operatorname{InverseGamme}(1/2,\, 1/a^2)
\end{align*}

References

  • High-Dimensional Bayesian Regularised Regression with the BayesReg Package (ArXiv)

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