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As far as I know, the Monte Carlo method may refer to:

  • Monte Carlo test, which means a method that estimates a value by random sampling;
  • Monte Carlo integration, which means a method that calculates the integration of a function by random sampling

However, the MCMC is simply a class of sampling method that uses the Markov Chain. I have no idea what does the word "Monte Carlo" means in this name.

If it is because:

  • "Mento Carlo" simply means "Sampling"?
  • or MCMC is often used for integration so this would be "Markov Chain (Sampling) and Mento Carlo (Integration)"?

I am now talking about why a sampling algorithm is called Monte Carlo, and I think there is aparent difference from this

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  • $\begingroup$ @StephanKolassa I have editted my question. $\endgroup$
    – C.K.
    Commented Oct 9, 2021 at 15:58
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    $\begingroup$ Monte Carlo is famous for its casinos. Casinos feature gambling. Gambling is the original inspiration for probability. (Not sure it’s necessary to overthink this.) $\endgroup$
    – Stephen Thomas
    Commented Oct 9, 2021 at 16:10
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    $\begingroup$ The linked thread answers your question. The name comes from casino. Commenting on your second bullet: MCMC is used to approximate integrals by sampling. It’s about sampling. $\endgroup$
    – Tim
    Commented Oct 9, 2021 at 18:29

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One common method is to formulate a Markov Chain, the limiting distribution of which is the posterior distribution for Bayesian inference. Then the chain is simulated through many steps to approximate the posterior distribution.

For one fairly simple example see the Gibbs Sampler section here.

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