10.4 ABC: Approximate Bayesian Computation

10.4.1 The ABC rejection algorithm

We sometimes find that we have a model with a likelihood which is mathematically difficult to deal with, and for such cases a technique called Approximate Bayesian Computation or ABC (sometimes referred to as likelihood-free computation) has been developed and has been quite widely employed, especially in genetics. There are several variants of this technique and we will illustrate each of them with the genetic linkage example we considered earlier in Sections 9.2 and 9.3.

Suppose we want to find the posterior distribution  when we have observations  coming from a distribution  and a (proper) prior distribution  for θ. Then if the observations come from a discrete distribution we can use the following algorithm to generate a sample of size k from the required posterior distribution. ABC algorithm

1. Generate θ from the prior (discrete) density  .
2. Generate from the density .
3. If , accept θ as an ...

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