from PSU astrostatistics lecture notes:

*MCMC is a form of simulation, because MCMC generates random numbers and bases the inference on them. However, there are two important differences between MCMC and the simulation that we shall discuss here.*

a) In MCMC we do not simulate from the target distribution directly. Instead, we simulate a Markov Chain that converges to the target distribution. Thus, MCMC performs approximate simulation.

b) The primary use of MCMC is in Bayesian statistics where we seek to simulate from the posterior distribution of the parameters. Classical simulation on the other hand simulates fresh data.a) In MCMC we do not simulate from the target distribution directly. Instead, we simulate a Markov Chain that converges to the target distribution. Thus, MCMC performs approximate simulation.

b) The primary use of MCMC is in Bayesian statistics where we seek to simulate from the posterior distribution of the parameters. Classical simulation on the other hand simulates fresh data.

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