¿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 forBayesians
the prior
is updated in the light of new data that is the evidence (Sober, 2008)
from this perspective you
could give answer to all questions.

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.

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