  MCMC algorithms offer empirically efficient tools for sampling from a target
distribution $\pi(x) \propto \exp(-V(x))$. However, on the theory side, MCMC
algorithms suffer from slow mixing rate when $\pi(x)$ is non-log-concave. Our
work examines this gap and shows that when Poincar\'e-style inequality holds on
a subset $\mathcal{X}$ of the state space, the conditional distribution of MCMC
iterates over $\mathcal{X}$ mixes fast to the true conditional distribution.
This fast mixing guarantee can hold in cases when global mixing is provably
slow. We formalize the statement and quantify the conditional mixing rate. We
further show that conditional mixing can have interesting implications for
sampling from mixtures of Gaussians, parameter estimation for Gaussian mixture
models and Gibbs-sampling with well-connected local minima.
