How's Monte Carlo different from random sampling with replacement?

Because Wikipedia page for Monte Carlo says:

Monte Carlo methods vary, but tend to follow a particular pattern:

  1. Define a domain of possible inputs
  2. Generate inputs randomly from a probability distribution over the domain
  3. Perform a deterministic computation on the inputs
  4. Aggregate the results

Isn't 2 about doing exactly the same thing as sampling with replacement?

Or is it Monte Carlo, because one does 3. and 4. in addition?


marked as duplicate by Xi'an, whuber Feb 7 '18 at 18:01

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    $\begingroup$ I think "Monte Carlo methods" is a general term "randomized algorithms". It includes a lot of algorithms. $\endgroup$ – Haitao Du Feb 7 '18 at 17:17
  • $\begingroup$ @hxd1011 In that case it's just very imprecise for lecturers to use the term Monte Carlo method in various cases, because it doesn't really tell what the method is used for. $\endgroup$ – mavavilj Feb 7 '18 at 17:34
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    $\begingroup$ Monte Carlo methods are a generic hat for evaluations involving simulation from a fixed distribution and it is not naturally associated with Statistics (or only with Statistics). Sampling with replacement seems connected with the bootstrap method, which means simulating from the empirical cdf of an iid sample. This is a fairly special case of Monte Carlo. $\endgroup$ – Xi'an Feb 7 '18 at 17:58
  • $\begingroup$ @mavavilj humans use many imprecise, general words. It is easier to say "flowers" then "a set containing roses, tulips, orchids, lilys, ..." don't you think? Monte Carlo is basically a synonym for simulation. $\endgroup$ – Tim Feb 7 '18 at 18:03
  • $\begingroup$ @Tim - I almost agree. There are creations such as flight simulators that create an experience, a set of sensory stimuli, rather than a set of quantitative findings to analyze. $\endgroup$ – rolando2 Feb 7 '18 at 21:00

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