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Comment by cf | Hacker News Reader

While I think detailed balance is typically how we explain why Metropolis Hastings works at all. It's actually not quite what matters. What matters is ergodicity and detailed balance is a proxy for checking for that. That's helpful since ergodicity does have an easier intuition behind it as well.

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Detailed balance is the specific way that ergodicity is guaranteed for Metropolis-Hastings (literally, it is the required aspect of the proof when you decompose the transition function into the proposal and acceptance / rejection functions), hence why detailed balance is the specific thing for this case.

I think we are agreeing. My point is you can get away with explaining Metropolis-Hastings in a useful way and never mention detailed balance. You will need it to work out a formal proof of convergence, but you don't need it to understand or even code up the algorithm.

I think explanations don't require formal proofs to be useful.

No, in the case of Metropolis-Hastings, you do need to understand detailed balance to understand the algorithm, because the conditions required of your proposal function literally come from it... not in a vague way about math convergence, but literally in the algorithm. For asymmetric proposal functions this is critical to understand.

Otherwise you are cargo cult copying some code and expecting it to work, without understanding the algorithm the code is executing.

Further you need to understand the convergence aspects too, because when you realize a real sample in a chain from your software application, whether or not you can safely use that chain of samples for the estimation or simulation you intended really depends on autocorrelation and convergence criteria. You cannot just believe the code was OK so the samples can be used... but “intuition” tutorials like this give a false sense of security that you can.

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