# What are the properties of a half Cauchy distribution?

I am currently working on a problem, where I need to develop a Markov chain Monte Carlo (MCMC) algorithm for a state space model.

To be able to solve the problem, I have been given the following probability of $\tau$: p($\tau$) = 2I($\tau$>0)/(1+$\tau^2$). $\tau$ being the standard deviation of $x$.

So now I know that it is a half-Cauchy distribution, because I recognise it from seeing examples and, because I was told so. But I do not fully understand why it is a "Half-Cauchy" distribution and which properties come with it.

In terms of properties I am not sure what I want. I am fairly new to this type of econometrics theory. So it is more for me to understand the distribution and how we use in a state space model context. The model itself looks like this: \begin{align} y_t &= x_t + e_t \\ x_{t+1} &= x_t + a_{t+1} \\[10pt] a_{t+1} &\sim ~ N(0, \tau^2) \\ p(\sigma^2) &\propto 1/\sigma^2 \\[3pt] p(\tau) &= \frac{2I(\tau>0)}{\pi(1+\tau^2)} \end{align}

Edit: I included $\pi$ in p($\tau$). Thank you for pointing this out.

• Please indicate which properties you are interested in: after all, there are infinitely many that one could describe. – whuber Oct 1 '16 at 2:54
• Half-distributions of symmetric distributions have twice the area functional height for their range, which range is a half-range, often but not necessarily starting at zero $x\geq 0$. – Carl Oct 1 '16 at 3:11

A half-Cauchy is one of the symmetric halves of the Cauchy distribution (if unspecified, it is the right half that's intended): Since the area of the right half of a Cauchy is $\frac12$ the density must then be doubled. Hence the 2 in your pdf (though it's missing a $\frac{1}{\pi}$ as whuber noted in comments).

The half-Cauchy has many properties; some are useful properties we may want in a prior.

A common choice for a prior on a scale parameter is the inverse gamma (not least, because it's conjugate for some familiar cases). When a weakly informative prior is desired, very small parameter values are used.

The half-Cauchy is quite heavy tailed and it, too, may be regarded as fairly weakly informative in some situations. Gelman ( for example) advocates for half-t priors (including the half-Cauchy) over the inverse gamma because they have better behavior for small parameter values but only regards it as wealy informative when a large scale parameter is used*. Gelman has focused more on the half-Cauchy in more recent years. The paper by Polson and Scott  gives additional reasons for choosing the half-Cauchy in particular.

* Your post shows a standard half-Cauchy. Gelman would probably not choose that for a prior. If you have no sense at all of the scale, it corresponds to saying that the scale is as likely to be above 1 as below 1 (which may be what you want) but it wouldn't fit with some of the things Gelman is arguing for.

 A. Gelman (2006),
"Prior distributions for variance parameters in hierarchical models"
Bayesian Analysis, Vol. 1, N. 3, pp. 515–533
http://www.stat.columbia.edu/~gelman/research/published/taumain.pdf

 N. G. Polson and J. G. Scott (2012),
"On the Half-Cauchy Prior for a Global Scale Parameter"
Bayesian Analysis, Vol. 7, No. 4, pp. 887-902
https://projecteuclid.org/euclid.ba/1354024466

• The "2" in the formula is superfluous, because apparently the formula is intended to give the PDF only up to a constant--after all, it is already missing the factor of $1/\pi$. – whuber Oct 1 '16 at 20:54
• @Glen_b, what is the location in the half-Cauchy in your answer? – rnorouzian Feb 15 '17 at 21:20
• @morouzian which location measure are you interested in? Regarded as a member of a location-scale family the standard form being discussed has a location of 0 and scale of 1, but I am not sure if that's what you're asking. (Its median is 1, as suggested near the end of my answer, if that helps any.) – Glen_b -Reinstate Monica Feb 15 '17 at 22:56