Weak Poincaré inequality comparisons for ideal and hybrid slice sampling

Using the framework of weak Poincaré inequalities, we provide a general comparison between Hybrid and Ideal Slice Sampling in terms of their corresponding Dirichlet forms. In particular, under suitable assumptions Hybrid Slice Sampling inherits fast convergence from Ideal Slice Sampling and conversely. We apply our results to analyse the convergence of the Independent Metropolis-Hastings, stepping-out and shrinkage, as well as Hit-and-Run within slice sampling algorithms.