Parsimony, Likelihood, Common
Causes, and Phylogenetic Inference
Elliott Sober
Philosophy Department
University of Wisconsin, Madison
1
2 suggested uses of O’s razor
• O’s razor should be used to constrain the
order in which hypotheses are to be
tested.
• O’s razor should be used to interpret the
acceptability/support of hypotheses that
have already been tested.
2
Pluralism about Ockham’s razor?
• [Pre-test] O’s razor should be used to
constrain the order in which hypotheses
are to be tested.
• [Post-test] O’s razor should be used to
interpret the acceptability/support of
hypotheses that have already been tested.
3
these can be compatible, but…
• [Pre-test] O’s razor should be used to constrain
the order in which hypotheses are to be tested.
• [Post-test] O’s razor should be used to interpret
the acceptability/support of hypotheses that
have already been tested.
If pre-test O’s razor is “rejectionist,” then
post-test O’s razor won’t have a point.
4
these can be compatible, but…
• [Pre-test] O’s razor should be used to constrain
the order in which hypotheses are to be tested.
• [Post-test] O’s razor should be used to interpret
the acceptability/support of hypotheses that
have already been tested.
If the pre-test idea involves testing hypotheses one
at time, then it views testing as noncontrastive.
5
within the post-test category of
support/plausibility ...
• Bayesianism – compute posterior probs.
• Likelihoodism – compare likelihoods.
• Frequentist model selection criteria like AIC –
estimate predictive accuracy.
6
I am a pluralist about these broad philosophies …
• Bayesianism – compute posterior probs
• Likelihoodism – compare likelihoods
• Frequentist model selection criteria like AIC – estimate predictive
accuracy.
7
I am a pluralist about these broad philosophies …
• Bayesianism – compute posterior probs
• Likelihoodism – compare likelihoods
• Frequentist model selection criteria like AIC – estimate predictive
accuracy.
Not that each is okay as a global thesis
about all scientific inference …
8
I am a pluralist about these broad philosophies …
• Bayesianism – compute posterior probs
• Likelihoodism – compare likelihoods
• Frequentist model selection criteria like AIC – estimate predictive
accuracy.
But I do think that each has its place.
9
Ockham’s Razors*
Different uses of O’s razor have different
justifications and some have none at all.
* “Let’s Razor Ockham’s Razor,” in D. Knowles (ed.), Explanation and
Its Limits, Cambridge University Press, 1990, 73-94.
10
Parsimony and Likelihood
In model selection criteria like AIC and BIC,
likelihood and parsimony are conflicting desiderata.
AIC(M) = log[Pr(Data│L(M)] - k
11
Parsimony and Likelihood
In model selection criteria like AIC and BIC,
likelihood and parsimony are conflicting desiderata.
In other settings, parsimony
has a likelihood justification.
12
the Law of Likelihood
Observation O favors H1 over H2
iff
Pr(O│H1) > Pr(O│H2)
13
a Reichenbachian idea
Salmon’s example of plagiarism
E1
E2
C
[Common Cause]
E1
E2
C1
C2
[Separate Causes]
14
a Reichenbachian idea
Salmon’s example of plagiarism
E1
E2
C
[Common Cause]
E1
E2
C1
C2
[Separate Causes]
15
more parsimonious
Reichenbach’s argument
IF
(i)
(ii)
(iii)
(iv)
A cause screens-off its effects from each other
All probabilities are non-extreme (≠ 0,1)
a particular parameterization of the CC and SC models
cause/effect relationships are “homogenous” across branches.
THEN
Pr[Data │Common Cause] > Pr[Data │Separate Causes].
16
parameters and homogeneity
E1
E2
p1
p2
C
[Common Cause]
E1
p1
C1
E2
p2
C2
[Separate Causes]
17
Reichenbach’s argument
IF
(i)
(ii)
(iii)
(iv)
A cause screens-off its effects from each other.
All probabilities are non-extreme
parameterization of the CC and SC models.
cause/effect relationships are “homogenous” across branches.
THEN
Pr[Data │Common Cause] > Pr[Data │Separate Causes].
The more parsimonious hypothesis
has the higher likelihood.
18
Reichenbach’s argument
IF
(i)
(ii)
(iii)
(iv)
A cause screens-off its effects from each other.
All probabilities are non-extreme
parameterization of the CC and SC models.
cause/effect relationships are “homogenous” across branches.
THEN
Pr[Data │Common Cause] > Pr[Data │Separate Causes].
Parsimony and likelihood are
ordinally equivalent.
19
Some differences with Reichenbach
• I am comparing two hypotheses.
• I’m not using R’s Principle of the Common Cause.
• I take the evidence to be the matching of the
students’ papers, not their “correlation.”
20
empirical foundations for
likelihood ≈ parsimony
(i)
(ii)
(iii)
(iv)
A cause screens-off its effects from each other.
All probabilities are non-extreme
parameterization of the CC and SC models.
cause/effect relationships are “homogenous” across branches.
By adopting different assumptions, you can
arrange for CC to be less likely than SC.
Now likelihood and parsimony conflict!
21
empirical foundations for
likelihood ≈ parsimony
(i)
(ii)
(iii)
(iv)
A cause screens-off its effects from each other.
All probabilities are non-extreme
parameterization of the CC and SC models.
cause/effect relationships are “homogenous” across branches.
Note: the R argument shows that these are
sufficient for likelihood ≈ parsimony,
not that they are necessary.
22
Parsimony in Phylogenetic
Inference
Two sources:
─ Willi Hennig
─ Luigi Cavalli-Sforza and Anthony Edwards
Two types of inference problem:
─ find the best tree “topology”
─ estimate character states of ancestors
23
1. Which tree topology is better?
H
C
(HC)G
G
H
C
G
H(CG)
MP: (HC)G is better supported than H(CG) by data D if and only if
(HC)G is a more parsimonious explanation of D than H(CG) is.
24
An Example of a Parsimony Calculation
1
H
1
C
0
G
1
H
1
C
0
0
(HC)G
H(CG)
0
G
25
2. What is the best estimate of the character states
of ancestors in an assumed tree?
1
H
1
C
1
G
A=?
26
2. What is the best estimate of the character states
of ancestors in an assumed tree?
1
H
1
C
1
G
A=?
MP says that the best estimate is that A=1.
27
Maximum Likelihood
H
C
(HC)G
G
H
C
G
H(CG)
ML: (HC)G is better supported than H(CG) by data D if and only if
Pr[D│(HC)G] > Pr[D│H(CG)].
28
Maximum Likelihood
H
C
(HC)G
G
H
C
G
H(CG)
ML: (HC)G is better supported than H(CG) by data D if and only if
PrM[D│(HC)G] > PrM[D│H(CG)]. ML is “model dependent.”
29
the present situation in evolutionary
biology
• MP and ML sometimes disagree.
• The standard criticism of MP is that it assumes
that evolution proceeds parsimoniously.
• The standard criticism of ML is that you need to
choose a model of the evolutionary process.
30
When do parsimony and likelihood
agree?
• (Ordinal Equivalence) For any data set D and any pair of
phylogenetic hypotheses H1and H2, Pr(D│H1) > Pr(D│H2) iff
H1 is a more parsimonious explanation of D than H2 is.
31
When do parsimony and likelihood
agree?
• (Ordinal Equivalence) For any data set D and any pair of
phylogenetic hypotheses H1and H2, PrM(D│H1) > PrM(D│H2)
iff H1 is a more parsimonious explanation of D than H2 is.
• Whether likelihood agrees with parsimony depends on the
probabilistic model of evolution used.
32
When do parsimony and likelihood
agree?
• (Ordinal Equivalence) For any data set D and any pair of
phylogenetic hypotheses H1and H2, PrM(D│H1) > PrM(D│H2)
iff H1 is a more parsimonious explanation of D than H2 is.
• Whether likelihood agrees with parsimony depends on the
probabilistic model of evolution used.
• Felsenstein (1973) showed that the postulate of very low
rates of evolution suffices for ordinal equivalence.
33
Does this mean that parsimony
assumes that rates are low?
• NO: the assumptions of a method are the
propositions that must be true if the method
correctly judges support.
34
Does this mean that parsimony
assumes that rates are low?
• NO: the assumptions of a method are the
propositions that must be true if the method
correctly judges support.
• Felsenstein showed that the postulate of low
rates suffices for ordinal equivalence, not that it
is necessary for ordinal equivalence.
35
Tuffley and Steel (1997)
• T&S showed that the postulate of “no-commonmechanism” also suffices for ordinal equivalence.
• “no-common-mechanism” means that each
character on each branch is subject to its
own drift process.
36
the two probability models of evolution
Felsenstein
Tuffley and Steel
• Rates of change are low,
but not necessarily equal.
• Rates of change can be
high.
• Drift not assumed:
Pr(i  j) and Pr(j  i)
may differ.
• Drift is assumed:
Pr(i  j) = Pr(j  i)
37
How to use likelihood to define what it means for parsimony
to assume something
• The assumptions of parsimony = the propositions that
must be true if parsimony correctly judges support.
• For a likelihoodist, parsimony correctly judges support if
and only if parsimony is ordinally equivalent with
likelihood.
• Hence, for a likelihoodist, parsimony assumes any
proposition that follows from ordinal equivalence.
38
A Test for what Parsimony does not
assume
Model M
ordinal equivalence
A
where A = what parsimony assumes
39
A Test for what Parsimony does not
assume
Model M
ordinal equivalence
A
where A = what parsimony assumes
• If model M entails ordinal equivalence, and M entails proposition X,
X may or may not be an assumption of parsimony.
40
A Test for what Parsimony does not
assume
Model M
ordinal equivalence
A
where A = what parsimony assumes
• If model M entails ordinal equivalence, and M entails proposition X,
X may or may not be an assumption of parsimony.
• If model M entails ordinal equivalence, and M does not entail
proposition X, then X is not an assumption of parsimony.
41
applications of the negative test
• T&S’s model does not entail that rates of change
are low; hence parsimony does not assume that
rates are low.
• F’s model does not assume neutral evolution;
hence parsimony does not assume neutrality.
42
How to figure out what parsimony
does assume?
• Find a model that forces parsimony and
likelihood to disagree about some
example.
• Then, if parsimony is right in what it says
about the example, the model must be
false.
43
Example #1
Task: Infer the character state of the MRCA of species that
all exhibit the same state of a quantitative character.
10
10 … 10
10
A=?
The MP estimate is A=10.
When is A=10 the ML estimate? And when is it not?
44
Answer
10
10 … 10
10
A=?
ML says that A=10 is the best estimate (and thus agrees with MP) if
there is neutral evolution or selection is pushing each lineage
towards a trait value of 10.
45
Answer
10
10 … 10
10
A=?
ML says that A=10 is the best estimate (and thus agrees with MP) if
there is neutral evolution or selection is pushing each lineage
towards a trait value of 10.
ML says that A=10 is not the best estimate (and thus disagrees with
MP) if (*) selection is pushing all lineages towards a single trait
value different from 10.
46
Answer
10
10 … 10
10
A=?
ML says that A=10 is the best estimate (and thus agrees with MP) if
there is neutral evolution or selection is pushing each lineage
towards a trait value of 10.
ML says that A=10 is not the best estimate (and thus disagrees with
MP) if (*) selection is pushing all lineages towards a single trait
value different from 10.
So: Parsimony assumes, in this problem, that (*) is false.
47
Example #2
Task: Infer the character state of the MRCA of two species that
exhibit different states of a dichotomous character.
1
0
A=?
A=0 and A=1 are equally parsimonious. When are they equally
likely? And when are they unequally likely?
48
Answer
1
0
A=?
ML agrees with MP that A=0 and A=1 are equally good estimates if the
same neutral process occurs in the two lineages.
49
Answer
1
0
A=?
ML agrees with MP that A=0 and A=1 are equally good estimates if the
same neutral process occurs in the two lineages.
ML disagrees with MP if (*) the same selection process occurs in both
lineages.
50
Answer
1
0
A=?
ML agrees with MP that A=0 and A=1 are equally good estimates if the
same neutral process occurs in the two lineages.
ML disagrees with MP if (*) the same selection process occurs in both
lineages.
So: Parsimony assumes, in this problem, that (*) is false.
51
Conclusions about
phylogenetic parsimony ≈ likelihood
• The assumptions of parsimony are the propositions that
must be true if parsimony correctly judges support.
52
Conclusions about
phylogenetic parsimony ≈ likelihood
• The assumptions of parsimony are the propositions that
must be true if parsimony correctly judges support.
• To find out what parsimony does not assume, use the
test described [M  ordinal equivalence  A].
53
Conclusions about
phylogenetic parsimony ≈ likelihood
• The assumptions of parsimony are the propositions that
must be true if parsimony correctly judges support.
• To find out what parsimony does not assume, use the
test described [M  ordinal equivalence  A].
• To find out what parsimony does assume, look for
examples in which parsimony and likelihood disagree,
not for models that ensure that they agree.
54
Conclusions about
phylogenetic parsimony ≈ likelihood
• The assumptions of parsimony are the propositions that
must be true if parsimony correctly judges support.
• To find out what parsimony does not assume, use the
test described [M  ordinal equivalence  A].
• To find out what parsimony does assume, look for
examples in which parsimony and likelihood disagree,
not for models that ensure that they agree.
• Maybe parsimony’s assumptions vary from problem to
problem.
55
broader conclusions
• underdetermination: O’s razor often comes up when the
data don’t settle truth/falsehood or acceptance/rejection.
56
broader conclusions
• underdetermination: O’s razor often comes up when the
data don’t settle truth/falsehood or acceptance/rejection.
• reductionism: when O’s razor has authority, it does so
because it reflects some other, more fundamental,
desideratum.
57
broader conclusions
• underdetermination: O’s razor often comes up when the
data don’t settle truth/falsehood or acceptance/rejection.
• reductionism: when O’s razor has authority, it does so
because it reflects some other, more fundamental,
desideratum. [But there isn’t a single global justification.]
58
broader conclusions
• underdetermination: O’s razor often comes up when the
data don’t settle truth and falsehood.
• reductionism: when O’s razor has authority, it does so
because it reflects some other, more fundamental,
desideratum.
• two questions: When parsimony has a precise meaning,
we can investigate: What are its presuppositions? What
suffices to justify it?
59
A curiosity: in the R argument, to get a difference
in likelihood, the hypotheses should not specify the
states of the causes.
E1
E2
p1
p2
C
[Common Cause]
E1
p1
C1
E2
p2
C2
[Separate Causes]
60
Example #0
Task: Infer the character state of the MRCA of species that
all exhibit the same state of a dichotomous character.
1
1 …
1
1
A=?
The MP inference is that A=1. When is A=1 the ML inference?
61
Example #0
Task: Infer the character state of the MRCA of species that
all exhibit the same state of a dichotomous character.
1
1 …
1
1
A=?
The MP inference is that A=1. When is A=1 the ML inference?
Answer: when lineages have finite duration and the process is Markovian.
It doesn’t matter whether selection or drift is the process at work.
62