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I am a little confused about the differences between Approximation Bayesian Computation (ABC) and Monte Carlo Methods (MCM). Citing from wikipedia:

Approximate Bayesian computation (ABC) constitutes a class of computational methods rooted in Bayesian statistics. In all model-based statistical inference, the likelihood function is of central importance, since it expresses the probability of the observed data under a particular statistical model, and thus quantifies the support data lend to particular values of parameters and to choices among different models. For simple models, an analytical formula for the likelihood function can typically be derived. However, for more complex models, an analytical formula might be elusive or the likelihood function might be computationally very costly to evaluate.

Monte Carlo methods (or Monte Carlo experiments) are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. [..] Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution.

In both cases, we run many simulation starting with constant or varying parameters and get the posterior probability (although I am not sure we call that a posterior probability in MCM). I would tend to think that ABC is MCM, where we use the posterior probability to define a confidence interval. Or maybe, by definition MCM are deterministic and therefore the resulting distribution is a function of the parameter values only and therefore it does not make any sense to make any statistical inference (such as calculating an confidence interval). I'd tend to think that ABC is a type of MCM, where we make statistical inference (or whenever the simulation takes a random seed in input). Is that right?

  • Is ABC a specific case of MCM?
  • Is ABC just like an MCM except that a random seed is taken in input?
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Yes, ABC is a specific application of Monte Carlo method. That application is approximating likelihood functions.

Anything that happens in a computer is actually deterministic. However, ABC, like any other MCM, must have a good pseudo-random generator. There is no difference in the amount of randomness required by the methods.

Examples of Monte Carlo methods which are not ABCs include genetic algorithms and Monte Carlo integration.

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