Does Approximate Bayesian Computation (ABC) follow the Likelihood Principle?

Question

I know that ABC is commonly used when the likelihood is intractable, so likelihood principle is not an interest in that case. But, I am curious whether the ABC satisfies the likelihood principle when the likelihood function is tractable. ABC is a generative procedure to sample parameters from posterior, and likelihood principle says that the inference on the parameter should be solely determined by likelihood part ignoring the term of the observation.

I think that if I generate fake samples from a parameter, the generating process is crucially affected by the term of observation, which might be ignored in the likelihood principle.

It's confusing, because I think that the ABC does not follow the likelihood principle, but it is well-known that Bayesian stat follows it.

Am I missing something?

Answer 1

The "when the likelihood function is tractable" is somewhat self-defeating, as the reason for using ABC is that it is intractable.

As for the likelihood principle, ABC is definitely not respecting it, since it requires a simulation of the data from its sampling distribution. It thus uses the frequentist properties of that distribution rather than the likelihood itself. Except in the (unrealistic) limiting case when the tolerance is exactly zero and the distance is based on the sufficient statistic, the ABC thus fails to agree with the likelihood principle.

In my humble opinion, this is a minor issue when compared with the major problems faced by ABC, unless you can provide an example with dire (There are also exact Bayesian approaches that do not agree with the likelihood principle, witness the Jeffreys or matching priors.)