A long run and a missed mode

Topic: Markov chains, Gibbs sampling, and Metropolis–Hastings · more questions on this topic
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Suppose the first B draws are badly influenced by initialisation. Compare their proportion in runs of length 100{,}000 and 100 billion. Then construct a two-mode example in which increasing the run length does not help on any practical time scale because the chain never crosses the low-density gap.

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