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Importance Sampling can estimated the normalizing constant by averaging the weights (the ratio of unnoramlized distribution and importance distribution). Is there anyway that MCMC algorithm can estimate the normalizing constant as well?

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    $\begingroup$ The most common way is Charles Geyers' reverse logistic regression method, which is published as a Tech Report here. $\endgroup$ – Greenparker Feb 7 at 9:57
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While I produced a detailed answer to an earlier X validated question, let me recall here that there are many ways of approximating the normalising constant, besides importance sampling:

  1. Chib's (1994) (or the candidate's) formula
  2. Gelfand and Dey's (1995) representation, which includes the infamous harmonic mean estimator
  3. particle filters and sequential Monte Carlo
  4. nested sampling
  5. reversible jump MCMC
  6. path sampling or thermodynamic integration
  7. bridge sampling
  8. Geyer's (1994) logistic regression
  9. the Savage-Dickey representation

some of which are unbiased and most of which are described in Chen, Shao and Ibrahim (2001).

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