Last time, I have introduced a parameterization of the half-Cauchy distribution as a hierarchical inverse gamma distributions that allows simple Gibbs sampler (Half-Cauchy distribution). I have just found another parameterization that also allows Gibbs sampler so I wanted to make a post about it.

So there are now two parameterizations of the half-Cauchy distribution:

In both cases, follows a half-Cauchy distribution. The second reparameterization, however, expressed as a ratio of gamma random variables, yields a very unstable Gibbs sampler. Consider a linear regression model

Let’s assign a horseshoe prior on $\beta_{j}$ which is expressed as a scale mixture of normal:

Then, we can set where and derive expressions for and instead. Then, the Gibbs samplers are constructed as follows:

The issue here is that is not defined but we can expect that sometimes irrelevant could degenerate to zero and in such cases, the sampler crashes. Perhaps, not use it? I don’t know how we can circumvent such issues.

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