The Likelihood Approach a.k.a. the 'third way' versus Bayesian

Question

In his book "In All Likelihood" Yudi Pawitan writes that "the likelihood approach offers a distinct 'third way', a Bayesian-frequentist compromise. We might call it Fisherian as it owes most of its conceptual development to Fisher (1890-1962). Fisher was clearly against the use of the axiomatic prior probability fundamental to the Bayesians, but he was equally emphatic in his rejection of long-run frequency as the only way to interpret probability. Fisher was a frequentist in his insistence that statistical inference should be objectively verifiable; however, his advocacy of likelihood inference in cases where probability-based inference is not available puts him closer to the Bayesian school".

How is the 'Likelihood Approach' different from the Bayesian approach?

Bayesian and Frequentism are very different philosophies and approaches, while the 'Likelihood' approach sounds like it is a twin of the Bayesian approach, with some very subtle difference somewhere... too subtle for me to understand at the moment. I haven't read Yudi Pawitan's book but I feel like he would disagree and would maintain that the 'Likelihood Approach' is very different than the Bayesian approach; it may differ subtly but that subtlety makes a world of a difference. I would love someone to explain this or correct me if I am getting this wrong.

One last question here is just a simple "Why do we not hear much about the 'Likelihood Approach'"? Certainly, not as much as we hear of Bayesian.

Answer 1

How is the 'Likelihood Approach' different from the Bayesian approach?

It's very different. Bayesians interpret probabilities in subjectivist terms as the measure of belief. They also use priors, so out-of-data knowledge, as parts of the models. Those are core concepts of the Bayesian approach, that is completely lacking from the likelihoodist approach.

Why do we not hear much about the 'Likelihood Approach'?

Because it blended with mainstream statistics. Maximum likelihood is one of the most popular ways of estimating model parameters, the likelihood ratio tests are commonly used. The discussion is usually about Bayesian vs non-Bayesian approaches because of the “controversial” ideas mentioned above. In the discussions, you would often see people using “frequentist” as a catch-all term for non-Bayesians.