Biomedical Engineering Reference
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consistency and the unclear dependence of the posterior density function on the
a priori information [
82
] possibly make the bayesian paradigm still unripe for
phylogenetic estimation [
1
].
8.5.2
Drawbacks of the Likelihood and the Bayesian Paradigms
of Phylogenetic Estimation
The higher the complexity of a paradigm, the higher the number of draw-backs
that could arise, and the likelihood and the bayesian paradigms do not escape the
rule. Specifically, a number of computational and theoretical drawbacks affect the
two paradigms. The computational drawbacks mainly involve (i) the optimization
aspects of the likelihood function and (ii) the sampling process in the bayesian
paradigm. The theoretical drawbacks concern the evolutionary hypotheses at the
core of the likelihood and bayesian criteria.
As regards to the computational drawbacks, in Sect.
8.5
we have seen that find-
ing the most likely phylogeny for a set of taxa involves maximizing a nonlinear and
generally non-convex stochastic function over all the possible phylogenies in
,
and for each phylogeny, over all the possible edge weights and substitution prob-
abilities. Notoriously, this task can be only performed in an approximate way, due
to a lack of general mathematical conditions that guarantee the global optimality
of a solution in nonlinear non-convex programming [
21
,
54
]. Hence, although it is
possible (at least for small datasets) to enumerate all the possible phylogenies in
T
,
it is not possible to optimize globally edge weights and the substitution probabilities
of a fixed phylogeny
T
. This fact may affect negatively the statistical consistency
of the likelihood and the bayesian paradigms. In fact, the local optima of the likeli-
hood function grows up exponentially in function of the number of taxa considered
[
7
,
19
,
20
]. Thus, fixed a phylogeny
T
, the global optimum of the likelihood function
is generally approximated by means of hill-climbing techniques that jump from lo-
cal optimum to another one until a stopping criterion is satisfied (e.g., the number of
iterations performed or the elapsed time) [
7
,
28
,
64
]. Assume that two phylogenies
T
T
1
2
be two vectors whose entries are edge weights
and the substitution probabilities associated to
T
2
1
and
are given, and let
and
T
1
T
2
, respectively. Let
z
1
and
and
T
1
and
T
2
for
1
and
2
, respectively, and assume, with-
z
2
, the likelihood values of
out loss of generality, that
z
1
>
z
2
. Due to the local nature of the optima
1
and
2
, there could exists another local optimum, say
2
, such that
O
z
2
>
z
1
>
z
2
.If
the hill-climbing algorithm finds
2
before
O
2
, then we will consider
T
2
as a better
phylogeny than
T
1
. Hence, it is easy
to realize that if one of the two phylogenies is the true phylogeny, its acceptance
is subordinated to the goodness of the hill-climbing algorithm used to optimize the
likelihood function, and as a result the statistical consistency of the likelihood and
bayesian paradigms may be seriously compromised.
Some authors argued that multiple local optima should arise infrequently in
real datasets [
64
], but this conjecture was proved false by Bryant et at. [
7
]and
T
1
, otherwise we will discard
T
2
in favor of
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