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During studying the convergence assessment on Markov Chain Monte Carlo, I once read the following statement:

A slowly converging sampler may be indistinguishable from one that will never converge(e.g., due to nonidentifiability)!

How to understand this statement? Thanks!

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It's sometimes very difficult to tell by looking at the various available convergence criteria whether a sampler is converging or not. If it's wandering around the space it may be converging slowly, or it may not be converging at all. Sometimes you can't tell the difference.

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What it means is that if each sample is 10.12345, 10.12345, 10.12345, 10.12345, 10.12345, 10.12345, 10.12345, 10.12345, for thousands of iterations, one might imagine that the chain has converged on 10.12345. (In practice, it would be oscillating around this value, but let's pretend the distribution has a tiny tiny variance for simplicity). Now, we might look at this and think that it's converged, but if we waited another 300 million iterations, maybe the chain would become: 10.12347, 10.12347, 10.12347. It's changed, slightly. And if we waited 10^30 iterations, for the sake of argument, maybe it would be 10.28152.

In this toy example, it's not that it had converged on 10.12345, it's just that the convergence rate was so slow as to have very little effect in the short term.

Like watching the hour hand of a clock, which appears stationary, but isn't really.

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