If we have a markov chain with the aim of generating a sample from some distribution $f(x)$, how can we diagnose whether the mixing of the chain is 'good' or 'bad'.

As I understand it, mixing is how fast the chain explores the support of $f(x)$.

So I think the time it takes for the chain to reach some stationary distribution is one indication. Is there any other diagnostic we can use by just visually inspecting the chain? Perhaps how frequently it gets stuck on successive values, etc?

I'm specifically talking within the context of Metropolis-Hastings algorithm.


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