Is Slice Sampling a special case of Gibbs Sampling? I read on this thread the following:

If you can use both the gibbs sampler and slice sampling to sample
  from a posterior I would use the Gibbs sampler as the slice sampler
  seems unnecessary to me <...> If you cannot use the gibbs but you can use the slice then your question seems irrelevant. 

But then, when on these slides from M. Jordan's lecture he says that:

Slice sampling is a special case of Gibbs sampling (in a product
  space).

What does he mean by that? 
Also, when would one not be able to do Gibbs sampling but still do slice sampling? 
 A: Slice sampling is a special case of Gibbs sampling, where one introduces a latent uniform random variable and then applies Gibbs to the extended set of rvs - this is what Jordan means by in a product space. Slice sampling can also be used within an existing Gibbs scheme, again by augmenting the parameter space (although one does not need to store the uniform parameters during sampling) so I guess what the answer in the other thread means is that it makes no sense to augment an existing Gibbs sampler with slice-within-Gibbs (EDIT: unless you can't run the existing Gibbs in the first place, of course). 
It's worth pointing out that there are generalizations of univariate slice sampling (e.g. reflective or elliptical) that can perform much better than Gibbs sampling, so I don't agree with the other answerer about never preferring slicing over Gibbs.
A: I disagree that slice sampling is strictly a special case of Gibbs sampling. It's almost Gibbs sampling, but not quite. Look at Neal (2003) section 3, which talks about sampling from $f(x)$ using an auxiliary variable $y$. A Gibbs sampler would


*

*Sample $y$ from $p(y | x)$ = Uniform(0, $f(x)$).

*Sample $x$ from $p(x | y)$ = Uniform($S$), where $S = \{x: y < f(x)\}$.


Step 1 is an exact Gibbs step, and if we knew the slice $S$ exactly, step 2 could be as well. However, we don't know $S$ in practice, so slice sampling uses successive approximations of $S$ instead of $S$ itself. Different ways to do this included the stepping-out slice sampler and the doubling slice sampler, both suggested by Neal. But in any case, step 2 is not an exact, true Gibbs step. 
I've read that slice sampling is a special case of a general auxiliary variable method, which relies on the same data augmentation idea: i.e., introducing an extra variable like $y$ and using it similarly to make sampling $x$ easier.
