An overview of the Weitzman approach to diversity

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An overview of the Weitzman approach to diversity

Caroline Thaon d’Arnoldi, Jean-Louis Foulley, Louis Ollivier

To cite this version:

Original

article

An

overview of

the

Weitzman

approach

to

_diversity

Caroline Thaon d’Arnoldi, Jean-Louis

_Foulley*

Louis Ollivier

Station de

_{génétique quantitative}

et

_appliquée,

Institut national de la recherche

_agronomique,

78352

_{Jouy-en-Josas cedex,}

France

(Received

10

July

1997;

accepted

30

_January

₁₉₉₈₎

Abstract - The

diversity

of a set of breeds or

_species

is defined in the Weitzman

approach

by

a recursion formula

_using

the

pairwise genetic

distances between the elements of the

set. The

_algorithm

for

_computing

the

_diversity

function of Weitzman is described. It also

provides

a _taxonomy of the set which is

_interpreted

as the maximum likelihood

_phylogeny.

The

_theory

is illustrated

_by

an

_application

to 19

_European

cattle breeds. The

_possible

uses of the method for

_defining

_optimal

conservation

_strategies

are

_briefly

discussed.

©

Inra/Elsevier,

Paris

diversity

/

taxonomy

/

conservation

_/

_phylogeny

_/

genetic distance

Résumé - Un aperçu sur _l’approche de la diversité selon Weitzman. La diversité d’un

ensemble

_{d’espèces,}

ou de _races, est définie _par Weitzman de

_{façon récursive ;}

les données de

_départ

sont les distances

_génétiques

entre les éléments de l’ensemble

_pris

deux à deux.

L’algorithme

de calcul de la diversité

_fournit,

comme résultat

_{intermédiaire,}

un arbre de

classement des

espèces

en

présence, qui

est

_interprété

comme une

phylogénie

du maximum

de vraisemblance. La théorie est illustrée par un

_{exemple d’application}

à 19 races bovines

européennes,

et les utilisations

possibles

de la méthode pour définir des

stratégies optimales

de conservation sont discutées brièvement.

_©

_{Inra/Elsevier,}

Paris

diversité

_/

taxonomie

_/

conservation

_/

_phylogénie

_/

distance _génétique

1. INTRODUCTION

The

_question

of

_preserving

_{biological diversity}

is

_currently

_attracting

a

_great

deal

of attention. Choices are _{necessary when it} comes to

_deciding

which

_endangered

species

must be

_protected

and which not.

_Conserving

breeds of farm

_animals,

or

domestic animal

_diversity,

_{presents strong}

_analogies

with the more

_general

_question

of

preserving

biological diversity.

In both cases,

owing

to the limited resources *

which can be devoted to

_{conservation,}

the central

_question

is ’what to

_preserve’

_!6!.

The choices are difficult and it would be much easier if an

operational

theoretical

framework based on this

_concept

of

_{’diversity’}

were available. As noted

_by

Solow

et al.

_!5!,

this

concept

of

_diversity

itself appears to have not so far been

precisely

defined,

apart

from a few

_attempts

which can be traced back to

_May

_!3!.

An

_analytical

framework able to

_guide

actual conservation

_policy

in a

diversity-improving

direction

_through

the use of a

_diversity

function has been

_{provided by}

Weitzman,

an

_economist,

who has

_given

an

_example

of

_application

to the

_problem

of crane

_species

conservation

!8-10!.

Since his

theory

is recent and almost unknown

to animal

_geneticists

_(see,

_however,

_Cunningham

_[1]

and Ollivier

_!4!),

and as it has

not

yet

been used in the context of livestock breed

_diversity,

we found it useful to

describe it

_{briefly and,}

as an

_{illustration,}

to

_apply

it to a set of cattle breeds.

2. THEORY

The method

_applies

to ’elements’ which _may

_represent

_{species, breeds, subspecies}

or _any other

operational

taxonomic unit. Pairwise distances between elements are

given, presenting

basic

_properties

of

_positivity,

_symmetry

and nil distance of an

element to itself. It is concerned with

_diversity

between

_units;

the

_theory

_ignores

diversity

due to variation within units.

2.1.

_Computing

_diversity

Computing

diversities is

_{straightforward}

if one knows how much the addition of one

_{element, say j,}

increases the

_diversity

of a

_given

set

_{Q. Intuitively,}

the

_magnitude

of the

_gain

should be related to how different the new element is from the set

_Q;

the more

_{different j}

is from

Q,

the

_greater

the

_gain.

This difference is measured

by

the distance

_d(j,

_Q).

_Here,

the distance from a

_{point j}

to a set

_Q

is

_defined,

as

usual in set

_{theory, by}

_min

_iEQ

_{d(i, j),}

in other

words,

the distance

_{between j}

and

its closest

_neighbour

in

_Q.

More

_precisely,

the intuitive

_property

of the

diversity

function

_(which

will be called V from now

on)

is the

_{’monotonicity}

in

species’:

the

_gain

of one element

increases the

_diversity

_by

at least

_{d(j, Q)}

However,

this is too loose a

_property

to define a

_unique

function. In

fact,

we will

consider

₍₁₎

as

general

conditions to

_satisfy

for any member i withdrawn from the

whole set

_S,

i.e.

where

_{B is}

the

_complement

set

_symbol,

i.e. here

_SBi

stands for S without i. Let

_V’

be defined as

V

i

’

=

V (SBi)

₊

d(i, SBi).

For _a

_given

_set

S,

the value of V’

If such a condition holds for the

_largest

V

i

’,

it will also be true for all the other

ones since:

According

to

_(2),

all the functions

_{having larger}

values than

_V’

also meet the

criterion;

to make the definition of

_V(S)

_unique,

it will be restricted to the lowest

one

_(minimum

of

_V),

i.e.

_precisely

to that

equal

to

_V’.

This leads to the recursive definition of the Weitzman

_diversity

function as:

with the initial conditions

The value of K is taken

_by

Weitzman

_{[8, 9]}

as a

_normalizing

constant which

computationally

can be set to zero.

Equation

(4)

provides

a

_unique

function

_having

some

_{interesting properties:}

- the ’twin

property’:

the addition of an element which is identical to an element

of S does not increase

_V;

- the

monotonicity

in

_species

_{[see (1)!;}

-

the

_continuity

in distances: if the

_pairwise

distances in set S are

slightly

modified,

the modification of

_diversity

is

_slight

_too;

- the

monotonicity

in distances: if every

pairwise

distance in set S is

_increased,

the

_diversity

of S increases too.

These

_properties

are fundamental.

They

have the merit to remove

_ambiguity

and to

_lay

down the definition of

_diversity

on

_simple

and

_rigorous

_principles.

In

particular,

the

_property

of

_continuity

in distances is of critical

_importance

for any

utilization of the

_{results, given}

that there is some

_uncertainty

on the real values of

the

_pairwise

distances.

2.2. The fundamental

_{representation}

theorem

The

_dynamic

_programming

recursion of

_equation

₍₄₎

involves n!

calculations,

n

being

the number of elements.

Fortunately,

the

following

_property

allows us to

reduce this

_computation

to 2! calculations. The

_dynamic

_programming

recursion

produces,

as a

_{secondary result,}

a

_graphical

_{representation}

of the relations between

the elements.

2.2.1. Link

_property

By definition,

and as shown

_previously,

there exists an element _{i in any} set S for

which the maximum of

_equation

₍₄₎

is achieved:

Weitzman has shown that the element i in

_{d(i, SBi)}

is one of the two closest

element i in S the loss of which involves a minimal reduction of

_{diversity equal}

to

d(i, SBi).

This element is called the link.

2.2.2. Theorem

Having

identified such a

_pair

(i, j),

how will we know which one is the link?

Remember from

₍₃₎

that

_V(S)

=

max (V’,

V! ).

Now V’

=

d(i, j)

₊

V(SBi),

and

Vj

=

d(i, j)+V (SB j)

_so that the link is the element

_satifying

_max

{V (SBi), V (SB j) }.

The

_{dynamic programming}

recursion becomes:

where, using

Weitzman’s

_notations,

the element

_g(S),

_satisfying

_{max [V(SBg),}

V(SBh)!

is called the

link,

the other one,

h(S),

is the

representative.

A

_proof

of the theorem can

_easily

be written

_by

mathematical induction with

respect

to the size of the set S.

2.2.3.

_Algorithm

and

_{graphical representation by}

a taxonomic tree

Applying

equation

(6)

recursively

generates

a rooted directed tree whose

twig-tips

are the elements of the set S and the nodes are the unknown ’ancestors’.

The different

_steps

of the

_algorithm

to be

_applied

_recursively

are

_(beginning

with the value of

_diversity

set to

_zero):

i)

find the two closest

_neighbours

_{i and j}

_among

the elements of S and add

_{d(i, j)}

to

_diversity;

ii)

determine the

_{link g}

and the

_{representative}

h

_by

_using

the

_property:

iii)

given

V(S)

=

d(g, h)

₊

V(SBg),

consider a new set without the link _g, i.e.

_SBg;

iv)

return to

_i)

until the size of the current set reaches

_1;

then add the constant

K defined in

₍₄₎

to

_diversity

and

_stop.

While

_drawing

the

_tree,

it is useful to

_place

the

_{link g}

between the

_{representative}

h and the closest

_neighbour

of h in

_QBg,

_Q

_being

the subset whose

_diversity

is

_computed

at this

_step.

_Intuitively,

it means that the loss of the link is less

consequential

for the

_diversity

than the loss of any other element. It

presents

the

advantage

of

_{allowing only}

one

_symmetry

_through

the

_{possible representations}

for

the

_tree,

while most hierarchical

clustering

methods result in a number of

possible

representations by

rotation of the branches. The

_diversity

of the set S can be read on the tree as the sum of the branch

_lengths,

or the sum of the ancestor ordinates. Weitzman also showed that the

_particular

tree

_generated

_by

the

_dynamic

recursion

_algorithm

in

₍₆₎

and

_steps

i-iv can be

_interpreted

as the tree

_maximizing

2.2.4.

_Example

Let us consider a set of four

_primate

_species.

Pairwise distances are

_given

in the

following

matrix

_(data

are

provided

by

Weitzman

!9!):

The closest

_neighbours

to be found in the set

_{Go,

_Or,

_HyL,

_HyS}

are

_HyL

and

HyS.

V{Go,

Or,

HyL,

HyS}

=

max [V{Go,

Or,

HyL}, V{Go,

Or,

HyS}]

₊

d(HyS, HyL)

Now we need to know which element is the link in the

couple

(HyL, HyS).

The

_following

matrices contain

pairwise

distances for the subsets

_{Go,

_Or,

_HyL}

and

_{Go,

_Or,

_HyS}:

V{Go,

Or,

HyL}

=

d(Go, Or) +max[V{Go,

HyL}, V{Or,

HyL}]

=

d(Go, Or)

₊

d(Go,

HyL) (so

Or is the link element in

{Go,

Or,

HyL})

= 889

V{Go,

Or,

HyS}

=

d(Go,

Or)

₊

max {V{Or, HyS}, V{Go, HyS}}

=

d(Go, Or)

₊

d(Go, HyS) (so

Or is the link element in

{Go,

Or,

HyS})

= 855

V{Go,

Or,

HyL}

>

_V{Go,

_Or,

_HyS},

thus we have determined that the link

element in the

_couple

_{(HyL, HyS)}

is

_HyS,

and

_consequently

the

_{representative}

is

HyL.

Considering

the

remaining

set after the

suppression

of the link

element,

i.e.

link element. This information then makes it

_possible

to

_compute

the total

_diversity,

which is worth 1015 =

d(Go, HyL)

₊

d(Go, Or)

₊

d(HyL, HyS),

and to draw the

corresponding

taxonomic tree

_{(figure 1).}

The link

_HyS

in

_{Go,

_Or,

_HyL,

_HyS}

is

_placed

between the

representative

HyL

and the closest

_neighbour

Or of

_HyL

in

_{Go,

_Or,

_HyL}.

The link Or in

_{Go,

_Or,

HyL}

is then

_placed

between the

representative

Go and the closest

_{neighbour HyL}

of Go in

_{{Go, HyL},}

_resulting

in a final order of

Go, Or, HyS, HyL.

3. APPLICATION: EXAMPLE OF EUROPEAN CATTLE BREEDS 3.1. Evaluation of

_diversity

The Weitzman method has been

applied

to data collected

_by

F. Grosclaude

[2]

on biochemical

_{polymorphisms}

(11

blood _group loci and the locus of blood serum transferrin and that of

beta-casein)

of 19

_European

cattle

_breeds,

_including

18 French breeds and the British Shorthorn. This latter was included because of

its Durham ancestor that has been introduced in some French

_regions

_during

the

last

_century.

The authors calculated the Nei standard distances

_considering

the 13

polymorphic

loci

_(table

_1).

Results of the different

_steps

of the

_computations

of

diversity

are shown in table II.

The

_graphical

_{representation}

of the result is shown in

_figure

2. A clear discrimi-nation is observed between two _groups i.e.

_i)

a first _group made of Northern

_dairy

breeds

_(Frisonne,

Flamande,

Maine

_Anjou,

_Shorthorn)

and

_ii)

another group

involv-ing

beef and

_hardy

breeds of the Center and West

_part

of France

_(Salers,

_Aubrac,

Limousine,

Charolais, Ferrandaise,

Blonde

_{d’Aquitaine)}

as well as Western and

Eastern dual _purpose breeds

_(e.g.

Pie

_{Rouge, Abondance,}

_Tarentaise,

Brune des

Current

_population

sizes in some of those breeds are so restricted that

they

are said to be

_endangered:

e.g. Bretonne Pie

Noire, Ferrandaise, Vosgienne

or the

Shorthorn.

The Weitzman method allows us to

_quantify

the loss of

_diversity

caused

_by

the extinction of _any subset _among the 19

_original

breeds.

_{By looking}

at the tree it is evident that the extinction of the Shorthorn causes a much

greater

loss of

diversity

than the extinction of the

_Flamande,

whose distance from its closest

_neighbour,

the Frisonne Pie

Noire,

is

_quite

small.

By

computing

the diversities of the initial set of breeds and the set minus the

Flamande,

or the

Shorthorn,

or both the Flamande and the

Shorthorn,

one finds that the loss of the set Flamande + Shorthorn induces a reduction of

diversity equal

to the sum of the reductions caused

_by

the loss of each of these breeds. This

_property

of

_additivity

is related to the

_degree

of

_{’independence’}

between the two breeds. On

The loss of

_diversity

caused

_by

the extinction of a set of breeds can be estimated

by

the sum of the ordinates of the nodes that would

_disappear

from the tree if the extinct breeds were to be

_removed,

without _any other

_change.

_{Thus, just by looking}

at the

tree,

it is obvious than the loss of the Normande would decrease the

diversity

eight

or nine times more than the loss of the Blonde

d’Aquitaine,

and even more

3.2. Further considerations on conservation

_strategies

The

_algorithm

may be

applied

to evaluate the relative merit of breeds with small

or medium

_population

sizes

_regarding

_diversity.

Let us consider the whole set

_(say

Q)

of the 18 French cattle breeds

_analysed

in this

_study,

and that

_{(say L)}

of the six

_{largest dairy}

_(Francaise

_Frisonne,

Montb6liarde and

_Normande)

and beef breeds

(Blonde

d’Aquitaine,

Charolaise and

_Limousine).

The relative loss due to

_keeping

those six breeds

_only

is 57.2

%.

Now one _{may ask} which is the most

_interesting

breed to select _among the rest if any of them has to be

_preserved.

This can be evaluated

by considering

the relative loss of

_diversity

between

_Q

and L

_plus

each of those 12

breeds. Results based on Nei and

(Cavalli-Sforza)

distances are the

following:

The breed

_providing

the lowest loss of

_diversity

is the Salers breed followed

_by

the Aubrac. The

_ranking

is consistent across the two distances used.

_Although

this

is

_only

an illustration which would deserve further

_{analysis including}

additional

markers,

this

example

is a

_significant

one as those breeds have been

_recognized

as

key hardy

breeds for a

long

time

[7].

4. DISCUSSION AND CONCLUSION

The method

_presented

provides

several results with different

_degrees

of robust-ness and different

_potential

_{applications.}

As indicated

_above,

the value of

_diversity

possesses a useful

_property

of

_continuity

in distances. The results _may be considered as relevant to

_support

decisions

_affecting

the breeds or

species

to be

_preserved.

The choice would be based

_only

on

objective

computations,

without

_relying

on such

subjective

characteristics as

_beauty,

interest

for future or

_present

_generations

or _any other intrinsic criterium.

_Experience

has

shown that it is difficult to base

_priorities

on such criteria.

The Weitzman

_approach

to

_diversity

allows further

_{developments.}

Weitzman

[10]

suggests

defining

a

_{diversity expected}

after a

_given

period

of

_time,

based on

the extinction

probability

of each element of the set considered. If n elements are

endangered,

2 survival-extinction

_patterns

_may occur with

_{given probabilities,}

and for each

_pattern

the

_{resulting diversity}

may be calculated. Weitzman then

defines a

_{’marginal diversity’}

of each

element,

obtained as the

_partial

derivative of the

_expected

_diversity

with

respect

to the extinction

_probability

of this element. The

_{marginal diversity}

of breed i measures the relative

_gain

in

_expected

_diversity

(after

50 years

say)

from

_improving

the survival

_probability

of breed i. In a similar

fashion,

one could assume that the extinction of a breed can be

_completely

avoided

by using

cryopreservation

and calculate the

_gain

in

_expected

_diversity

obtained

and the risk status of a

_given

set of

_endangered

breeds as

_expressed

_through

their

respective

probabilities

of

_extinction,

an order of

_priority

for a

_{cryopreservation}

programme could thus be established.

Because

_diversity

is

_{computed recursively,}

it involves very

long

calculations when the size n of the set is

_larger

than 25. The

_{approximation}

_proposed

in this

_study

relies on a random choice of the link at each

_stage

of the recursive

_algorithm,

i.e.

on

_sampling

trees among the

2

n

-

1

possible

trees. The

_procedure

can be

_applied

as follows:

_i)

compute

V among the elements of S

by choosing

at each

step

the link not from the formula in

_(6),

but at random out of the

_pair

of closest

_neighbours,

ii)

repeat

i)

m times such as to

_generate

m different values of

V,

iii)

take as the estimated value of

_V(S)

the maximum value of V over all values

computed.

This can be

_performed

_{by choosing}

at random m

_integers

smaller than

2!!! ,

convert

them into their

_binary

expression

and use the convention that the link will be the

first element if the value is 0 and the second if it is 1. This

_procedure

was tested on a set of 29 cattle breeds

_using

data from Moazami-Goudarzi

_{(pers. comm.).}

For

m = 10

_000,

the estimated value of V _{was at} least of 13 200 _as

compared

_{to a} real

value of 13

_722,

i.e. bias lower than 4

%.

This

_{approximation}

is

_quite

_{good regarding}

the time of

_computation

_{required by}

this estimation

_{(20 min)}

while the

_complete

algorithm

needed more than 8

days.

On the other

_hand,

the

_graphical

_{representation}

_might

be sensitive to

_slight

modifications of the distance matrix if the values of

_diversity

are close for

cer-tain subsets. Simulation

_procedures

to evaluate the robustness of clades have been

proposed by

Weitzman

_[8].

_Although

the

_clustering

_power looks

_satisfying

on the

examples

we

considered,

_any

_phylogenetic

_{interpretation}

of the results should be

used with caution. It should also be

_emphasized

that the use made of

_genetic

dis-tances in this

approach

differs from their use in

deriving genealogical

trees.

_Though

trees are useful

_{geometric representations}

of

diversity -

the

diversity

function de-fined above is indeed

_equal

to the total branch

_length

of the

_{corresponding}

tree

-they

must be considered as

_telling

the

evolutionary

_story

that best fits the

diversity

observed,

but not

_necessarily

as

_telling

the ’true’

_story.

In

_fact,

as

_{emphasized by}

Weitzman

_[9],

there is no need for the elements to have been

_{generated by}

_any

real

_{evolutionary phylogeny.}

This has to be

_kept

in mind

_particularly

when sets of domestic breeds are considered. Given the

_exchanges

known to have occurred in their

past

histories,

domestic breeds are indeed not

_likely

to have resulted from a

strict tree-like

_branching

process. Whereas taxonomists are

_essentially

interested in

finding

the

_evolutionary

_story

behind a

_given

observed

_diversity,

_{conservationists,}

especially

breed

conservationists,

do not need that

type

of information as

they

are more concerned with the future evolution of

_diversity.

The main use of the Weitzman method is to determine

_{preservation strategies.}

It

supposes,

however,

that the elements of the set considered are and remain distinct.

If this constraint can be

_removed,

it _may be

_suggested

that certain

endangered

breeds be

_amalgamated

with other ones. The

population

size would

_increase,

no

additional costs would be

_engaged,

and the direct loss of alleles that results from

an extinction could be avoided. Of _course, this

implies

that the breed standards

should be relaxed for a

while,

but it is a

dynamic

_conception

of

_preservation

that

Despite

the criticisms which can be raised

_against

the Weitzman

_approach,

including

that it

ignores

the differences in within unit

variation,

it should be

_kept

in mind that it does

_satisfy

certain basic

_properties

which do not

_always

hold with traditional criteria. The

_principle

₍₁₎

of

_{’monotonicity}

in

_species’

means that the

change

in

_diversity

_{V(SBi) - V(S)}

due to the loss of some

_population

i is

_always

negative

or nil

(for

i being

a twin

element).

In

_contrast,

this

_property

does not

apply

to

_variance,

for it can be

_easily

shown that the total variance of a mixture of

populations

can increase after some of them are deleted.

ACKNOWLEDGMENTS

This work was conducted while Caroline Thaon was on a

_’stage

de

fin

d’6tudes’ at the Station de

_{génétique quantitative}

et

_appliqu6e

_(SGC!A),

Inra, Jouy-en-Josas

as a student

from the Ecole

Polytechnique,

Palaiseau. She

_{greatly acknowledges}

the support of both

institutions in

_making

this stay feasible.

Special

thanks are

_expressed

to F. Grosclaude and K. Moazami-Goudarzi

_(Laboratoire

de

_{génétique biochimique,}

_{Jouy-en-Josas)}

for

providing

the data on cattle

_analysed

in this

_study.

We are also

_grateful

to C. Dillmann and P. Dubreuil

_(Inra,

Station de

_{génétique vegetale,}

Le

_Moulon)

and S. Lemarié

(Inra-SERD,

Grenoble)

for

_{having provided}

additional

_{test-examples,}

and to Bruce

_Southey

and

an _anonymous referee for their valuable comments which

_helped

to

_improve

the manuscrit. E.

_Thompson

is also thanked for her

English

revision of the text.

REFERENCES

[1]

Cunningham P.,

Genetic

diversity

in domestic animals:

strategies

for conservation

and

_development,

in: Miller

_R.H.,

Pursel

_V.G.,

Norman H.D.

_(Eds.),

XX

_{Biotechnology’s}

Role in the Genetic

_Improvement

of Farm

_Animals,

American

_Society

of Animal

_Science,

Savoy,

IL,

USA, 1996,

pp. 13-23.

[2]

Grosclaude

F., Aupetit R.Y.,

Lefebvre

_J.,

Mériaux

_J.C.,

Essai

_d’analyse

des relations

génétiques

entre les races bovines

_frangaises

à 1’aide du

_{polymorphisme biochimique,}

Genet. Sel. Evol. 22

₍₁₉₉₀₎

317-338.

[3]

May R.M., Taxonomy

as

_destiny,

Nature 347

(1990)

129-130.

[4]

Ollivier

_L.,

Génétique

et conservation

_animales,

in: Matassino

_{D., Boyazoglu J.,}

Capuccio

A.

_(Eds.),

International

Symposium

on Mediterranean Animal

Germplasm

and Future Human

_Challenges,

EAAP

publication

no.

85, Wageningen Pers, Wageningen, 1997,

pp. 211-219.

[5]

Solow

_{A., Polasky S.,}

Broadus

_J.,

On the measurement of

_{biological diversity,}

J. Environ. Econom.

Manag.

24

₍₁₉₉₃₎

60-68.

[6]

Vane-Wright R.I., Humphries C.J.,

Williams _P.H., What to

_{protect? Systematics}

and the agony of

choice,

Biol. Cons. 55

(1991)

235-254.

[7]

Vissac B., Etude

_génétique

de la race

_d’Aubrac,

in:

_{L’Aubrac, CNRS, Paris, I, 1970,}

pp. 29-102.

[8]

Weitzman

M.,

A reduced form

_approach

to maximum likelihood estimation of

evolutionary

trees, Harvard Institute of Economic

_{Research, Paper}

No.

_1569,

1991.

[9]

Weitzman

_M.,

On

_{diversity, Quart.}

J. Econ. 107

₍₁₉₉₂₎

363-405.

[10]

Weitzman

_M.,

What to

_preserve?

An

_application

of

_{diversity theory}

to crane

APPENDIX: the maximum likelihood tree

Weitzman

_[8]

_provides

the

_{following phylogenetic}

_{interpretation.}

Let us note

p(i, j)

the conditional

_probability

_{P(i! j)}

that a

_species

i exists

_given

that a

_species

j

exists. Assume that this

_probability

is a function of the

_genetic

distance between

i and

_j.

The

_{hypothesis underlying}

this

_assumption

is that the distance

_{d(i, j)}

between two

_species

_{i and j}

measures the time since their

_separation.

More

_precisely,

we will _suppose that

p(i, j)

=

exp

!-ad(i, j)]

where A is a ’universal extinction rate’.

The maximum likelihood tree is the evolution scheme

_(i.e.

the set of unknown

ancestors)

which maximizes the

_probability

that every element of S exists at the

current time.

Let

_{P( j !i)}

be the conditional

_probability

that

_{species j exists given}

i exists.

Assuming

that the evolution scheme is

_known,

it can be shown

_that,

for _any subset

Q

E

_S,

and J E

_SBQ,

the conditional

_probability

_P(jlQ)

_{that j survived}

_given

_Q

exists satisfies

Note

_p(j,

_Q)

=

m! P(j!i).

Now,

from basic

_{probability theory,}

_P(jlQ)

-t€Q

P(Q U j)/P(Q),

and

_combining

this with

_(A.1)

leads to:

Let us note

_11(8),

the

_{largest probability}

that S

_exists,

i.e. the

_probability

of existence under the most favourable evolution scheme.

_Equation

_(A.2)

applied

for

Q

=

SBi,

and j

₌

i implies

Any

evolution scheme that would induce a value of

P(S)

= II* would be identified as the scheme under which the

_probability

that S exists is

_maximal,

ie the maximum likelihood tree.

_Taking

the

logarithm

of

_equation

_(A.3)

and

_{normalizing A}

to

_1,

it

becomes:

Since

_(A.5)

has been studied above and solved

_{by algorithm}

_(6),

we are able to exhibit such an evolution scheme. The tree

_{generated by}

the Weitzman method