¿Bayesianism or Likelihoodism?
Let me start with the Royall's three questions:
1. ¿What does the present evidence say?
2. ¿What should you believe?
3. ¿What should you do?
Although Likelihoodists and Bayesians both share the likelihood principle and the law of likelihood which are important in the philosophy of scientific method, they disagree on several instances:
Its necessary highlight that the most remarkable difference between them is that Bayesians use prior probabilities in other words posterior probability distributions that require prior probability distributions and likelihood functions and likelihoodists not.
Another difference points to the meaning of evidence: Likelihoodists characterize data as evidence and they don't use them to guide our beliefs or actions and maintain that this characterization is valuable in itself (Royall 1997, Ch. 1). Then, you couldn't give answer to the second nor the third question of Royall because they say nothing about what you should believe after you receiving the evidence without take into account what you believe before receiving the evidence. On the other hand for
Regarding to the second question about your degree of belief Bayesians answer this question from the concept of confirmation where the observation (O) provides confirmation of hypothesis 1 (H1) when this has a higher likelihood than its own negation (Gandenberger, 2013). Unlike Likelihoodists whom doesn't use this concept of confirmation, they don't take into account if the evidence raises, lowers or not change the probability of the hypothesis. They compare hypotheses to each other which have their own likelihoods and use the law of likelihood to interpret the data where: the observation (O) favors hypothesis 1 (H1) over hypothesis 2 (H2) if Pr (O | H1) > Pr (O | H2) and the likelihood ratio is used to show the degree to which O favors H1 over H2 that is given by Pr(O | H1) / Pr(O | H2), and they ask if H1 has a higher likelihood than H2. So, for likelihoodists is enough use the likelihood ratio as a measure of degree favoring one hypothesis over other one (Sober, 2008). In contrast to Bayesian for whom is not enough and then implemented the use of posterior probabilities, see below.
In Bayesian inference you assign a probability to the hypothesis (H) before doing an observation in other words is the distribution of the parameters before doing analysis of the data (prior probability) and after of doing it there is a reallocation of the probability assigned to H and the probability in the light of evidence is known as posterior probability and is denoted Pr(H|O) that means probability of the Hypothesis given the Observation. In contrast Maximum likelihood where the likelihood of the hypothesis is the probability that H confers on O Pr(O|H) (Sober, 2008).
Other thing in common is that ML and BI use the same models of evolution, but the way to measure the support of relationships in the topology are different, ML uses bootstrap support (BS) which is a measure of confidence, and uses data resampling to estimate the support (Cummings et al., 2003). Unlike BI that uses the posterior probability (PP) which is calculated from prior probability, likelihood functions and data. Both measures have been controversial because of several reasons and some claim there is a equivalence between both measures (Efron, H. and Holmes, 1996), but some studies like Erixon et al. (2003) reject this assumption and others claim PP is a better measure of support (Alfaro, Zoller, and Lutzoni, 2003).
Given the similarities and differences between them I think that Bayesian inference is the best method of all.
Alfaro, M. E., Zoller, S., & Lutzoni, F. (2003). Bayes or bootstrap? A simulation study comparing the performance of Bayesian Markov chain Monte Carlo sampling and bootstrapping in assessing phylogenetic confidence. Molecular Biology and Evolution, 20(2), 255–266.
Cummings, M. P., Handley, S. A., Myers, D. S., Reed, D. L., Rokas, A., & Winka, K. (2003). Comparing bootstrap and posterior probability values in the four-taxon case. Systematic Biology, 52(4), 477–487.
Efron, B., Halloran, E., & Holmes, S. (1996). Bootstrap confidence levels for phylogenetic trees. Proceedings of the National Academy of Sciences, 93(23), 13429.
Erixon, P., Svennblad, B., Britton, T., & Oxelman, B. (2003). Reliability of Bayesian posterior probabilities and bootstrap frequencies in phylogenetics. Systematic Biology, 52(5), 665–673.
Gandenberger Greg . 2013. Why I am not a likelihoodist.
Royall, R. Statistical Evidence: A Likelihood Paradigm, Boca Raton, Fla.:Chapman and Hall.(1997).
SOBER, Elliott. Evidence and evolution: The logic behind the science. Cambridge University Press, 2008.