# How to sample random variables (x,y) from a bivariate Cauchy distribution using a Gibbs sampler?

A bivariate Cauchy distribution is equivalent to a bivariate t-distribution with 1 degree of freedom.

• Why not generate Cauchy variates directly? Is this homework? – Neil G Jan 9 '16 at 17:43
• It's a question from a past exam. – ori06 Jan 9 '16 at 18:03
• I can think of using the method of composition using the fact that the t-distribution can be written in terms of the ratio of a standard normal distribution and a chi-squared distribution. I can't figure how to do Gibbs sampling for the same problem... – ori06 Jan 9 '16 at 18:06
• Using the ratio is not Gibbs sampling. For Gibbs sampling, you only need the density of the Cauchy distribution and some proposal distribution. Do you know how Gibbs sampling works? – Neil G Jan 9 '16 at 18:36
• I know only the basics, that you sample at each step using the full conditional distributions i.e you sample $$x_i^{t+1}$$ from $$p(x_i|y, x_2^t,... x_n^t)$$ for i: 1 to n. How does the proposal distribution come into play for Gibbs sampling? I don't know anything about that... – ori06 Jan 9 '16 at 19:40