What does "mixing" mean in sampling? I keep seeing this term "mixing": when people want to show their sampler works better, they say it "mixes" better. The term is a little counter-intuitive.
 A: When people say "mixing" in the context of Markov chain Monte Carlo (MCMC), they are (knowingly or unknowingly) referring to the "mixing time" of the Markov chain.
Intuitively, mixing time for a Markov chain is the number of steps required of the Markov chain to come close to the stationary distribution (or in the world of Bayesian statistics, posterior distribution). If $\pi$ is the stationary distribution and $P(x,A)$ is the Markov chain transition kernel, where $x$ is the starting value of the Markov chain, and $A$ is a measurable set, then the mixing time is the first time $t$ such that
$$\left|P^t(x,A) - \pi(A)\right|_{TV} \leq \dfrac{1}{4}. $$
Here $|\cdot|_{TV}$ refers to total variation distance. This is only one of the many definitions, but they all intuitively mean the same.
The mixing time has a direct impact on sampling quality since, the smaller the mixing time, the faster the convergence of the Markov chain to the stationary distribution, and the smaller the correlation in the samples.
A: Well, if I remember correctly what little I ever knew about ergodic theory, mixing implies ergodic, and ergodic means that time averages and space averages are the same, which is the justification for sampling over a sequence of random samples. So one would like to know that one's sampling scheme is mixing.
I suppose this is really only an issue if one is inventing a new sampling scheme. But more often, one is just using some software off the shelf, with (one hopes) known properties.
