The goal of both methods seems to be to derive an estimate of a posterior/target distribution. If a process model exists which links some input parameters (which are themselves uncertain and can be described by a PDF) to an output parameter through a model equation or other computations, why would one choose one method over the other? Would both be applicable? Can one make a statement on the benefit of one method over the other with respect to the number of required draws/simulation runs in order to reach a sufficiently good approximation of the target PDF?
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2 Answers
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The short answer is: An MCMC is a MC, but not all MCs are MCMC.
The slightly longer answer: MC methods are a class of methods, of which MCMC is one possibility. Even MCMC does not uniquely define your method as there are different variations of MCMC.
You can read more in: Robert, C. P., & Casella, G. (2004). Monte Carlo statistical methods. New York: Springer.
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$\begingroup$ Note that the book referenced by Maarten is a classic but not an introductory book. It is quite focused on MCMC rather than simple MC and is rather a difficult read for beginners. $\endgroup$– M. ToyaCommented Jul 10, 2013 at 19:17
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$\begingroup$ @AlfredM. that is a fair characterization of that book $\endgroup$ Commented Jul 11, 2013 at 14:26
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1$\begingroup$ Fair enough!, but we also wrote an introduction to Monte Carlo methods with R! $\endgroup$– Xi'anCommented Apr 4, 2016 at 11:44
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1$\begingroup$ This answer could really benefit from a short note on what is is that makes MCMC a subset of MC. My understanding is that in MCMC subsequent samples are dependent on previous samples (via some function), where MC can include independent sampling. Not sure if that's the only difference though. $\endgroup$ Commented Feb 16, 2022 at 2:04
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In general Monte Carlo (MC) refers to estimating an integral by using random sampling to avoid curse of dimensionality problem. Also, once you have the samples, it's possible to compute the expectations of any random variable with respect to the sampled distribution.
A subclass of MC is MCMC you set up a Markov chain whose stationary distribution is the target distribution that you want to sample from. The main thing about many MCMC methods is that due to the fact that you've set up a Markov chain, the samples are positively correlated and thereby increases the variance of your integral/expectation estimates. The better situation is to make your samples independent (or have carefully constructed negative correlation) to reduce the variance. However, many distributions that you want to sample from are incredibly complicated objects and are difficult to sample from directly. Hence the construction and use of MCMC.
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$\begingroup$ As to positive correlation of samples, I agree but would note that it's not variables correlated with one another that's of primary concern; it's a given variable, correlated with itself, over time. Metropolis algorithm is by far the easiest MCMC sampler to implement, however, it's most vulnerable to auto-correlation over time. One of the most effective improvements upon it is HMC, which uses Hamiltonian dynamics to slingshot across the PDF to similar likelihood regions. When a large change in the parameter results in a small change in likelihood, auto-correlation is meaningfully reduced. $\endgroup$ Commented Apr 9, 2021 at 16:03
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$\begingroup$ This answer could do with a definition of what a Markov Chain is. $\endgroup$ Commented Feb 16, 2022 at 2:05