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I'm not sure why one would use ABC/Likelihood-free inference methods instead of standard Bayesian inference methods. Is this fundamentally a conceptual problem of mine?

Are there any concrete examples which could elaborate on when to use the former as opposed to straightforward Bayesian inference?

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Quoting the great Wikipedia article on ABC (emphasis added):

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.

ABC methods bypass the evaluation of the likelihood function. (...)

When using ABC we approximate the likelihood function using some kind of summary statistics, so you rather would not use it instead of non-approximate Bayesian computation. We can use it in situations, where we cannot use the non-approximate methods.

Check the famous socks example by Rasmus Bååth for some friendly introduction (see also here).

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