Two-scale criteria for Poincaré and log-Sobolev inequalities with applications to Markov chain Monte Carlo

Given a collection of distributions $\{P_{y}\}$ and a mixing distribution $ρ$ supported over $\mathbb{R}^{d}$, we propose new sufficient conditions under which the mixture / joint distribution satisfies a Poincaré or log-Sobolev inequality. We develop these sufficient conditions in a unified manner