Bayesian inference about dispersion parameters of univariate mixed models with maternal effects: theoretical considerations

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Bayesian inference about dispersion parameters of

univariate mixed models with maternal effects:

theoretical considerations

Rjc Cantet, Rl Fernando, D Gianola

To cite this version:

Original

article

Bayesian

inference about

_dispersion

parameters

of

univariate mixed models

with maternal effects:

theoretical considerations

RJC Cantet

RL Fernando, D

Gianola

University of Illanoas, Department of Animal Sciences, Urbana,

IL _61801, USA

(Received

14 _{January 1991; accepted} 5 _January

₁₉₉₂₎

Summary - Mixed linear models for maternal effects include fixed and random elements, and _dispersion parameters

(variances

and

_{covariances).}

In this paper a _Bayesian model

for inferences about such parameters is _presented. The model includes a normal likelihood

for the data, a "flat" prior for the fixed effects and a multivariate normal _prior for the direct and maternal

_breeding

values. The _prior distribution for the _genetic variance-covariance components is in the inverted Wishart form and the environmental _components follow

_inverted

_X

₂

_prior distributions. The kernel of the _{joint posterior density} of the dispersion parameters is derived in closed form. Additional numerical and _analytical methods of interest that are

_suggested

to _complete a _{Bayesian analysis} include Monte-Carlo _Integration, maximum entropy fit, asymptotic approximations, and the Tierney-Kadane _approach to _{marginalization.}

maternal effect _/ Bayesian method _/ dispersion parameter

Résumé - Inférence _bayésienne des _paramètres de _dispersion de modèles mixtes uni-variates avec effets maternels : considérations théoriques. Les modèles linéaires mixtes

avec _effets maternels _comprennent des éléments _fixés et _aléatoires, et des _paramètres de dis-persion

(variances

et

_{covariances).}

Dans cet article est

_présenté

un modèle

_6ayésien

_pour l’estimation de ces _paramètres. Le modèle _comprend une vraisemblance normale _pour les données, un a _{priori uniforme pour} les

_{effets fixés}

et un a _priori multivariate normal _pour les valeurs _génétiques directes et maternelles. Là distribution a _priori des _composantes de variance-covariance est une distribution de Wishart inverse et les _composantes de milieu

*

Correspondence and _{reprints; present} address: Facultad de

_Agronomia,

Universidad de Buenos _{Aires, Departamento} de _Zootecnia, 1417 Buenos _Aires,

_Argentina.

**

suivent des distributions a _priori de

_x

₂

inverse.

Le noyau de la densité conjointe a posteri-ori des _paramètres de _dispersion est _explicité. En outre, des méthodes _numériques et

ana-lytiques

sont

_proposées

pour compléter

l’analyse

bayésienne:

intégration

par des méthodes

de

_Monte-Carlo,

_ajustement par le maximum

d’entropie,

approximations asymptotiques et la méthode de _{marginalisation} de

_{Tiemey-Kadane.}

effet maternel _/ méthode bayésienne / paramètre de dispersion

INTRODUCTION

Mixed linear models for the

_study

of

_quantitative

traits

_include,

in addition to fixed and random

_effects,

_{the necessary}

_dispersion

_parameters.

_Suppose

one is interested

in

_making

inferences about variance and covariance components.

Except

in trivial

cases, it is

impossible

to derive the exact

_sampling

distribution of estimators of these _parameters

_{(Searle, 1979)}

so, at

_best,

one has to resort to

_asymptotic

results.

Theory

(Cramer,

1986)

indicates that the

_joint

distribution of maximum likelihood

estimators of several _parameters is

_{asymptotically normal,}

and therefore so are their

marginal

distributions.

_However,

this _may not

_provide

an

_adequate

description

of

the distribution of estimators with finite

_sample

sizes. On the other

hand,

the

Bayesian approach

is

_capable

of

_producing

exact

_joint

and

_marginal

_posterior

distributions for _any

_sample

size

_(Zellner,

_1971;

Box and

_Tiao,

_1973),

which

_give

a

full

_description

of the state of

_{uncertainty posterior}

to data.

In recent _years,

_Bayesian

methods have been

_developed

for variance

_component

estimation in animal

_{breeding (Gianola}

and

_{Fernando, 1986;}

_Foulley

et

_al,

_1987;

Macedo and

_Gianola,

_1987;

_Carriquiry,

_1989;

Gianola et

_{al 1990a,}

_b).

All these studies found

_analytically

intractable

_{joint posterior}

distributions of

_(co)variance

components, as

_Broemeling

(1985)

has also observed. Further

_{marginalization}

with

respect to

_dispersion

_parameters seems difficult or

_{impossible by analytical}

means.

However,

there are at least 3 other

_options

for the

_study

of

_marginal

_posterior

distributions:

_1),

_{approximations;}

_2),

_{integration by}

numerical _means; and

_3),

numerical

_integration

for

_computing

moments followed

_by

a fit of the

_density

using

these

_numerically

obtained

_{expectations.}

Recent advances in

_computing

have

encouraged

the use of numerical methods in

_Bayesian

inference. For

_example,

after the

_pioneering

work of Kloek and Van

_Dijk

_(1978),

Monte Carlo

_integration

(Hammersley

and

Handscomb, 1964; Rubinstein,

1981)

has been

_employed

in econometric models

_(Bauwens,

1984;

Zellner et

al,

1988),

seemingly

unrelated

regressions

(Richard

and

_Steel,

₁₉₈₈₎

and

_binary

responses

(Zellner

and

Rossi,

1984).

Maternal effects are an

_important

source of

_genetic

and environmental variation in mammalian

_species

_{(Falconer, 1981).}

Biometrical _aspects of the associated

theory

were first

_{developed by}

Dickerson

_(1947),

and

quantitative

genetic

models

were

proposed

by Kempthorne

(1955),

Willham

(1963, 1972)

and Falconer

(1965).

Evolutionary biologists

have also become interested in maternal effects

_(Cheverud,

Henderson,

1984,

1988).

Although

there are maternal sources of variation within and among

breeds,

we are concerned here

_only

with the former sources.

The _purpose of this

_expository

_{paper is} to _present a

_Bayesian

model for inference about variance and covariance _components in a mixed linear model

_describing

a trait affected

_by

maternal effects. The formulation is

_general

in the sense

that it can be

_applied

to the case where maternal effects are absent. The

_joint

posterior

distribution of the

_dispersion

_parameters is derived. Numerical methods for

_integration

of

_dispersion

_parameters

_regarded

as &dquo;nuisances&dquo; in

_specific

_settings

are reviewed.

_Among

_these,

Monte Carlo

_{integration by}

_{&dquo;importance}

_{sampling&dquo;}

(Hammersley

and

Handscomb, 1964; Rubinstein,

1981)

is discussed.

_Also,

_fitting

a &dquo;maximum

_{entropy&dquo;}

_posterior

distribution

_(Jaynes,

1957,

1979)

using

moments

obtained

_by

numerical means

(Mead

and

Papanicolaou,

_1984;

Zellner and

Highfield,

1988)

is considered.

_Suggestions

on some

_{approximations}

to

_marginal

_posterior

distributions of the

_(co)variance

_components are

_given.

_Asymptotic

_{approximations}

using

the

_Laplace

method for

_integrals

_(Tierney

and

_Kadane,

₁₉₈₆₎

are also

described as a means for

_obtaining

_{approximate posterior}

moments and

_marginal

densities. Extension of the methods studied here to deal with

_multiple

traits is

possible

but the

_algebra

is more involved.

THE BAYESIAN MODEL

Model and

_{prior assumptions}

about location

_parameters

The maternal animal model

_{(Henderson, 1988)}

considered is:

where _y is an n x 1 vector of records and

_X,

,

o

Z

Z

m

and

E

m

are

known,

fixed,

n x _p, n x _a, n x a and n x d

_matrices,

_{respectively;}

without loss of

_generality,

the matrix X is assumed to have full-column rank. The vectors

_!,

_{ao, am} and Cm are unknown fixed

_effects,

additive direct

_breeding

_values,

additive maternal

breeding

values and maternal environmental

_deviations,

_{respectively.}

The n x 1

vector eo contains environmental deviations as well as _any

_discrepancy

between the &dquo;structure&dquo; of the model

_(XR+

Z

o

a

o

+

_a

_m

_m

_Z

+

_E

_m

_e

_m

₎

and the data y. As in

Gianola et al

_(1990b),

the vectors

_p,a

_o

_,

am and em are

_formally

viewed as location

parameters of the conditional distribution

_{yl P,}

_ao, , maem, but a distinction is made

between 13

and the other 3 vectors

_depending

on the state of

_{uncertainty prior}

to

observing

data. It is assumed a

piiori

that P

follows,a

uniform

distribution,

so as to

reflect vague

prior

knowledge

on this vector.

_Polygenic

inheritance is often assumed for a =

[a!, a!]’

(Falconer,

_{1981; Bulmer,}

1985)

_so it is reasonable _to

postulate

a

prio

7

i

that a follows the multivariate normal distribution:

where G is a 2 x 2 matrix with

_diagonal

elements

_{o, Ao 2}

and

_{aA 2&dquo;&dquo;}

the variance

components for additive direct and maternal

_{genetic effects,}

_{respectively,}

and off

effects. The a x a

_{positive-definite}

matrix A has elements

_equal

to

_Wright’s

coefficients of additive

_relationship

or twice Melecot’s coefficients of _co-ancestry

(Willham, 1963).

Maternal environmental

_{deviations, presumably}

caused

_by

the

joint

action of many factors

having

relatively

small effects are also assumed to be

normally, independently

distributed

_(Quaas

and

_Pollak,

1980;

Henderson,

1988)

as:

where

_u5!

is the maternal environmental variance. It is assumed that a

_priori

_p,

a and Cm are

mutually independent.

For the _{vector y,} it will be assumed that:

where

_(1’!o

_is

the variance of the direct environmental effects. It should be noted that

[1-4J

_complete

the

_{specification}

of the classical mixed linear model

_(Henderson,

1984),

but in the

_latter,

distributions

_[2]

and

_[3]

have a

_frequentist

_{interpretation.}

A

_{simplifying assumption}

made in this

_model,

for

_analytical

_reasons, is that the direct and maternal environmental effects are uncorrelated.

Prior

_assumptions

about variance

_parameters

Variance and covariance _components, the main focus of this

_study,

_appear in the

distributions of a, emand eo. Often these components are unknown. In the

_Bayesian

approach,

a

_{joint prior}

distribution must be

_specified

for

_these,

so as to reflect

uncertainty prior

to

_observing

y. &dquo;Flat&dquo;

prior

distributions,

although

leading

to

inferences that are

_equivalent

to those obtained from likelihood in certain

_settings

(Harville,

1974,

1977)

can cause

_problems

in others

(Lindley

and

Smith,

_1972;

Thompson,

1980;

Gianola et

_al,

_1990b).

In this

_study,

informative

_priors

of the

type of _proper

_conjugate

distributions

_(Raiffa

and

_Schlaiffer,

₁₉₆₁₎

are used. A

prior

distribution is said to be

_conjugate

if the

_posterior

distribution is also in the same

family

For

_example,

a normal

_prior

combined with a normal likelihood

_produces

a

normal

_posterior

_(Zellner,

_1971;

Box and

_Tiao,

_1973).

_However,

as shown later for

the variance-covariance structure under

_{consideration,}

the

_posterior

distribution of the

_dispersion

_parameters is not of the same _type as their

_{joint prior}

distribution. This was also found

_by

Macedo and Gianola

₍₁₉₈₇₎

and

_by

Gianola et al

_(1990b)

who studied a mixed linear model with several variance _components

_employing

normal-gamma conjugate

prior

distributions.

An inverted-Wishart distribution

_(Zellner,

_1971;

_Anderson,

_1984;

_Foulley

et

_al,

1987)

will be used for

_G,

with

_density:

where G* =

_!c9Gh.

The 2 x 2 matrix Gh of

_{&dquo;hyperparameters&dquo;, interpretable}

as

prior

values of the

_dispersion

_parameters, has

_diagonal

elements

_s2

₀

and

_s

₂

_M

_,

and

values _{may be} difficult _{in many}

_{applications.}

Gianola et al

_(1990b)

_{suggested fitting}

the distribution to _past estimates of the

_(co)variance

_components

_by

_{eg a method} of moments fit. For traits such as birth and

_weaning

_weight

in cattle there is

a considerable number of estimates of the necessary

(co)variance

components in

the literature

_(Cantet

et

_al,

_1988).

_Clearly,

the value of

_G

_h

influences

_posterior

inferences unless the

_prior

distribution is overwhelmed

_by

the likelihood function

(Box

and

_Tiao,

_1973).

Similarly,

as in Hoeschele et al

₍₁₉₈₇₎

the inverted

_x

₂

distribution

_(a

_particular

case of the inverted Wishart

_{distribution)}

is

_suggested

for the environmental

variance components, and the densities are:

The

_prior

variances

_s2

_m

and

_{s 20}

are the scalar _counterparts of

_G

_n

_,

and no and nm are the

_{corresponding degrees}

of belief. The

_marginal

distribution of any

diagonal

element of a Wishart random matrix is

X

2

(Anderson,

1984).

_Likewise,

the

_marginal

distribution of the

_diagonal

of an inverted-Wishart random matrix

is inverted

_X

_Z

_{(Zellner, 1971).}

Note that the 2 variances in

_[6]

and

_[7]

cannot be

arranged

in matrix form similar to the additive

_(co)variance

_components in G to obtain an inverted Wishart

_density,

unless no = _n,n.

Setting

ng I n o

and nm to zero makes the

_prior

distributions for all

_(co)variance

_components

_{&dquo;uniformative&dquo;,}

in

the sense of Zellner

_(1971).

POSTERIOR DENSITIES

Joint

_{posterior density}

of all

_parameters

To facilitate

_{marginalization}

of

_[8],

and

_as

in Gianola et al

_(1990a),

let

W =

[XIZ

oI

Z

mI

E

m]

,

0’ =

[j

f

[a’[e£

]

and

define

i

such that

where the p + 2a + d square matrix E is

given by:

Using

this,

one can write:

Gianola et al

_(1990a)

noted that

in

_(9J

can be

_interpreted

as a &dquo;mixed model residual sum of

squares&dquo;. Using

[9]

in

Posterior

_density

of the

_(co)variance

_components

To obtain the

_marginal

_posterior

distribution of

_{G, u5!}

and

_O’!o,

0 must be

integrated

out of

_(10).

This can be

_accomplished

_noting

that the second

_exponential

term in

_[10]

is the kernel of the

_(p

+ 2a +

_d)-variate

normal distribution

and the variance-covariance matrix is

_non-singular

because X has full-column rank. The

_remaining

terms in

_[10]

do not

_depend

on 0.

Therefore,

with

_R

_o

_being

the _range of

_{0, using}

_properties

of the normal distribution we have:

The

_marginal

_posterior

distribution of all

_(co)variance

_components then is:

The structure of

_[11]

makes it difficult or

impossible

to obtain

_by

_analytical

means the

_{marginal posterior}

distribution of

_G,

_{o,2 . E}

or

or2 E,,,.

_Therefore,

in order to

make

_{marginal posterior}

inferences about the elements of G or the environmental

variances, approximations

or numerical

_integration

must be used. The latter _may

give

accurate estimates of

_posterior

_moments, but in

_{multiparameter}

situations

computations

can be

_prohibitive.

There are 2 basic

_approaches

to numerical

_integration

in

_{Bayesian analysis.}

The first one is based on classical methods such as

quadrature

(Naylor

and

Smith,

1982,

1988;

Wright, 1986).

Increased _power of _computers has made Monte Carlo numerical

_integration

_(MCI),

the second

approach,

feasible in

_posterior

inferences

in econometric models

_(Kloek

and Van

_Dijk,

_1978;

_Bauwens,

_1984;

Bauwens and

_{Richard, 1985;}

Zellner et

_al,

₁₉₈₈₎

and in other models

_(Zellner

and

_Rossi,

1984;

Geweke,

1988;

Richard and

_Steel,

_1988).

In MCI the error is

_inversely

proportional

to

_N

_l/2

_,

where N is the number of

_points

where the

_integrand

is

evaluated

_(Hammersley

and

_{Handscomb, 1964; Rubinstein,}

_1981).

Even

_though

this

_{&dquo;convergence&dquo;}

of the error to zero is not

_rapid,

neither the

_{dimensionality}

of the

_{integration region}

nor the

_degree

of smoothness of the function evaluated enter

POSTERIOR MOMENTS VIA MONTE CARLO INTEGRATION

Consider

_finding

moments of parameters

having

the

_{joint posterior}

distribution with

_density

_[11].

Let

r’ =

_{[or2 A ,, or2 A}

_M

0’ AoAm,

,

m

2

E

0’

or E

2

.]

,

and let

g(r)

be either

a

_scalar,

vector or matrix function of r of which we would like to _compute its

posterior

expectation. Also,

let

_(11!

be

_represented

as

p(T ! y, H),

where H stands for

_{hyperparameters.}

Then:

assuming

the

_integrals

in

_[12]

exist.

Different

_techniques

can be used with MCI to achieve reasonable accuracy. An

ap-pealing

one for

_computing

_posterior

moments

_(Kloek

and Van

_{Dijk, 1978; Bauwens,}

1984,

Zellner and

_Rossi,

_1984;

Richard and

_Steel,

₁₉₈₈₎

is called

_{&dquo;importance}

sam-pling&dquo;

(Hammersley

and

_Handscomb,

_1964;

_Rubinstein,

_1981).

Let

_I(r)

be a known

probability density

function defined on the space of

T;

I(r)

is called the

importance

sampling

function.

_Following

Kloek and Van

_Dijk

₍₁₉₇₈₎

let

_M(r)

be:

with

_[13]

defined in the

_region

where

_7(F)

> 0. Then

_[12]

is

_expressible

as:

where the

_expectation

is taken with _respect to the

_{importance density}

_I(r).

Using

a

standard

Monte Carlo

procedure

(Hammersley

and

_{Handscomb, 1964;}

Rubinstein,

1981),

values of r are drawn at random from the distribution with

density

I(r).

Then the function

_M(r)

is evaluated for each drawn value of

_r,

r

j

(i

=

_{1, ... ,}

m)

_say. For

sufficiently

large

m:

The critical

_point

is the choice of the

_density

function

_7(F).

The closer

_I(r)

is to

p(r ! y, H),

the smaller is the variance of

_M(r),

and the number of

_drawings

needed

to obtain a certain accuracy

(Hammerley

and

Handscomb,

1964;

Rubinstein,

1981).

Another

_important

_requirement

is that random

_drawings

of r should be

_relatively

simple

to obtain from

_{7(F) (Kloek}

and Van

_Dijk,

1978; Bauwens,

1984).

For location

parameters, the multivariate

_normal,

multivariate and matric-variate t and

_poly-t

distributions have been used as

_importance

functions

_(Kloek

and Van

_Dijk,

_1978;

Bauwens,

1984;

Bauwens and

_Richard,

_1985;

Richard and

_{Steel, 1988;}

Zellner et

from the inverted Wishart distribution. There are several

_problems

_yet to be solved and the

_procedure

is still

experimental

(Richard

and

Steel,

1988).

However,

results

obtained so far make MCI

_{by importance}

sampling

_promising

(Bauwens,

1984;

Zellner and

Rossi, 1984;

Richard and

_Steel,

_1988;

Zellner et

_al,

_1988).

Consider

_calculating

the mean of

_G,

o, 20

and

a

2

m

with

_{joint posterior density}

as

_given

in

_[11].

From

_[13]

and

_[14]:

Let now:

where:

7

i

(r)

=

prior

density

of G

((5!

times

k

l

,

the

integration

constant),

I

2

(r)

=

_prior

_density

of

U2

E

((6!

_times

_k

₂

_,

the

_integration

constant),

1

3

(r)

=

_prior

density

of u5!

[7]

times

k

3

,

the

_integration

constant).

Then:

where

k

o

is the constant of

_integration

of

_(11!.

_Evaluating

_{E(r ! y,}

_H)

then entails the

_following

_steps:

a)

draw at random the elements of r from distributions with densities

_{h (r)}

(inverted Wishart),

2

(r)

1

(inverted

x

2

)

and

_{(r) (inverted}

_I

₃

_X

_Z

_).

This can be done

using,

for

_example,

the

_algorithm

of Bauwens

_(1984).

b)

Evaluate

k

o

=

(J

[11] dT!-1.

Now,

Note that

_M

_o

is

_[18]

without r. Then

_k

_o

can be evaluated

_by

MCI

_{by computing}

the average of

M

o

,

and

_taking

its

_reciprocal.

c)

Once

M

o

is

_evaluated,

then _compute

_M(r)

= r

M

o

.

In order to

perform

_steps

(b)

and

_(c),

the mixed model

_equations

and the determinant of W’W + E need to

be solved and

_{evaluated, repeatedly,}

for each

_drawing.

The mixed model

_equations

can be solved

iteratively

and

_{diagonalization}

or _sparse matrix factorization

(Misztal,

1990)

can be

employed

to

_advantage

in the calculation of the determinant.

This

_procedure

can be used to calculate _any function of r. For

_example,

the

posterior

variance-covariance matrix is:

MAXIMUM ENTROPY FIT OF MARGINAL POSTERIOR DENSITIES

A full

_{Bayesian analysis}

requires

finding

the

_{marginal posterior}

distribution of each of the

_(co)variance

_components.

_Probability

statements and

_highest

_posterior

density

intervals are obtained from these distributions

(Zellner,

1971;

Box and

Tiao,

1973).

Marginal posterior

densities can be obtained

_using

the Monte Carlo method

(Kloek

and Van

_Dijk,

₁₉₇₈₎

but it is

_{computationally expensive.}

An alternative is

to _compute

_by

MCI some moments

_(for

_instance,

the first

₄₎

of each _parameter,

and then fit a function that

_approximates

the necessary

marginal

distribution. A

method that

_gives

a reasonable

fit,

&dquo;Maximum

entropy&dquo;

(ME),

has been used

by

Mead and

_Papanicolaou

₍₁₉₈₄₎

and Zellner and

_Highfield

_{(1988). Choosing}

the

ME distribution means

_assuming

the &dquo;least&dquo;

possible

(Jaynes, 1979),

ie,

using

information one has but not

_using

what one does not have. An ME fit based on the

first 4 moments

_{implies constructing}

a distribution that does not use information

beyond

that

_{conveyed by}

these moments.

_Jaynes

₍₁₉₅₇₎

set the basis for what is

known as the &dquo;ME formalism&dquo; and found a role for this to

_play

in

_Bayesian

statistics.

The _entropy

_(W)

of a continuous distribution with

density

p(x)

is defined

(Shannon,

1948; Jaynes,

1957,

1979)

to be:

The ME distribution is obtained from the

_density

that maximizes

_[20]

_subject

to t’he conditions:

r

where po = 1

(by

definition of a _proper

_density

function)

and

JLi

(i

=

1, ... , 4)

_are the first 4 moments of the distribution of x. Zellner and

Highfield (1988)

expressed

the function to be maximized as the

_Lagrangian:

where the

_l

_i

_(i

=

0,...,4)

_are

_{Lagrange multipliers}

and I =

!lo, ll, d2, d3, l4!’.

Note

formula

_[23]

is

_expressible

as:

Using

Euler’s

_equation

_{(Hildebrand, 1972)}

the condition for a

_stationary

_point

is:

Because H does not

_depend

on

p’(x), [25]

holds

_only

if

aH/ap(x)

=

_{0, ie,}

if:

Hence,

the condition for a

_{stationary point}

is:

plus

the 5 constraints

_given

in

_(21!.

From

_(26!,

the

_density

of the ME distribution of x has the form:

To

_specify

the ME distribution

_completely

1 must be found. Zellner and

_Highfield

(1988)

suggested

a numerical solution based on Newton’s method.

_Using

[27]

the side conditions

_[21]

can be written as:

These derivatives are

_simply

moments

_(with

_negative

_sign)

of the maximum

entropy distribution.

Putting

in

_[29]

and

_setting

this

_equal

to

_[28]

one obtains the linear _system in 1:

This _system can be solved for

_{h ( j}

=

0,1, ... , 4)

to obtain a new set of trial values

and,

thus,

an iteration is established.

_Defining

and

_observing

that

_{0 <_}

i

_{+ j <_}

_8,

the above _system can be written in matrix notation as:

This _system is solved for

8

1

’

l

to obtain

IN

=

Ilt-11

₊

1

ft]

,

the _vector of new

trial values. Iteration continues until 5 becomes

_{appropriately}

small. Zellner and

Highfield

(1988)

showed that coefficient matrix in

_[30]

is

_{positive definite,}

so solutions are

_unique.

In summary, the method includes 3 _types of

_{computations.}

First,

the moments _{pi - p4}must be

_computed

_by

some method such as

_MCI;

this is

done

_only

once.

Second,

the

G

i

values

(i

=

0,1, ... , 8)

_are

computed

at _every round of iteration

_carrying

out unidimensional

_{integrations,}

as indicated in

_[28].

Third,

the 5 x

_{5 system}

_[30]

is solved. At _{convergence, the ME}

_density

_[27]

is

_employed

to

SOME ANALYTICAL APPROXIMATIONS TO MARGINAL POSTERIOR DENSITIES

Because numerical

_integration

can be

_{computationally expensive}

and the _accuracy of MCI in this _type of

_problem

is still

_unknown,

we consider several

_{approximations}

to

_marginal

_posterior

distributions.

The mode of the

_{posterior density}

_[11]

can be found

by maximizing

this

jointly

with _respect to

_G,

_{o- E}

_2m

and

_u5!.

_Foulley

et al

_(1987),

Gianola et al

_(1990b)

and

Macedo and Gianola

₍₁₉₈₇₎

showed how this could be done with

_a

_’

_{simpte algorithm}

based on first derivatives. Additional

_algorithms

can be

constructed using

second

derivatives,

and the necessary

expression

are

_given

in the

_Appen.dix.

The solutions can be viewed as

weighted

averages of REML &dquo;estimators&dquo; of

dispersion

parameters

and

of the

_{hyperparameters}

_G

_h

_,

_sE

_o

and

_si

_m’

Let the modal values so obtained be

1i,

!2

and

_!2

or i’,

in _compact.

’

.°

&2 E.

and

_{&2 E ’n,}

_{or r,}

in _compact.

-Consider

_{approximations}

to the

_marginal

_density

of G because this matrix

contains the _parameters of

_primary

interests. One can write: ,

where

_{p(u5!, u5!}

_{y, H)}

is the

posterior

density

of

_{u5! , u5!}

_obtained after

inte-grating

G out of

_[11].

It seems

_impossible

to _{carry out} this

_{integration analytically.}

Following

ideas in Gianola and Fernando

_(1986),

we _propose as first

_{approximation:}

It would be better to use the modal values

of p(u5! , u 5!

y, H)

rather than

(TJ.:m

and

_3!,

but

_finding

this distribution does not seem feasible.

_Using

[32]

in

[11]

one obtains:

It should be noted that now

6

=

f (G, &’ E &dquo;&dquo; a2 E !)

and

t

=

h(G, a2 E &dquo;&dquo; 1?2 E ).

Then,

the MCI method can be used to _compute moments of

_(33J.

The additional

degree

of

_{marginalization}

with _respect to

_[11]

achieved in this

_{approximation}

may

be

_small,

but

_savings

in

_computing

accrue because

drawing

values of

uk

m

and

o-E

o

from

_I

₂

_(r)

and

_I

₃

_(r),

_{respectively,}

is no

_longer

_necessary.

In the

_{preceding, replace}

and

_using

the

_{preceding developments}

in

_[33]

we _write, after

neglecting

IW

I

W+EI-1/

2

This

_density

is in the inverted Wishart

form,

with _parameters

_n.

=

ng + a and

G*,

provided

G*

is

_positive

definite. If _not, one can &dquo;bend&dquo; this matrix

_following

the ideas

_{of Hayes}

and Hill

_(1981).

The

_{computational}

_advantage

of

_[34]

over

_[33]

is that

_{y’y -}

9 W’y

would be evaluated

_only

once at

G,Ô’k

m

,Ô’k

o

.

_Further,

the inverted Wishart form of

_[34]

_yields

an

_analytical

solution for the

(approximate)

marginal

posterior

densities of

_QAo

_and

_{o,2 A M,}

so

_approximate

_probability

statements

about elements of G can be made with relative ease. A third

_{approximation}

would be

_writing

_[34]

as

so we would have an inverted Wishart distribution with

hyperparameters

n9

₌

n

9 + a and G. If G is obtained with an

algorithm

that _guarantees

positive

semi-definiteness such as EM

(Dempster

et

_al,

_1977),

this would circumvent the

_potential

problem posed by G*

in

_(34!.

The fourth

_{approximation}

involves the matrix of second derivatives

_{(C, say)}

of the

_logarithm

of

_[11]

_with

_respect to the

_unique

elements of

_{G, u5!}

and

_o,

₂

_E

_o

and then

_evaluating

C at

1i, %5!

_and

_%5!.

The second derivatives are in the

Appendi!. Invoking

the

_asymptotic

_normality

property of

_posterior

distributions

(Zellner, 1971),

one would

approximately

have :

where it is assumed that the matrix -C =

f(l#, %5! , %5!)

has full rank. The

THE TIERNEY-KADANE APPROXIMATIONS

The

_{approximation}

in

_[36]

_produces

reasonable results when the

_posterior

distri-bution is

_unimodal,

which holds for

_large

_{enough samples. Tierney}

and Kadane

(1986)

described another

_{approximation}

_(based

on

_Laplace’s

method for

_integrals),

and this is reviewed in the

_following

section.

Single

parameter situation

Let

_g(r)

= g be a

_positive

function of the scalar _parameter r. Then

where l is the likelihood

_function,

₇

_r

is the

_prior

_density

and c is the

_integration

constant

-With n

_{being sample}

_size,

let

Using

[44]

in

_[37]:

The method of

_Tierney

and Kadane

₍₁₉₈₆₎

continues as follows. Let

r&dquo;!

be the

posterior

mode

_(which

is also the maximum of

_{L), L’(r)}

and

_L&dquo;(r)

be the first and second derivatives of L with _respect toT and let (1&dquo;2

_{= -1/ LI/ (r}

_m

₎

_’

_Using

a

_Taylor

series

_expansion

for

_nL(r)

about

_r&dquo;

_L

we have:

Noting

that

and

_retaining

terms up to

_{second-order,}

the

_expansion

becomes:

Using

this,

the denominator in

_[45]

can be

_approximated

as:

Taking

the ratio between

_[47]

and

_[46]

as

required

in

[45]

then,

approximately,

we have:

An

_interesting

_aspect of this

_{approximation}

is that

_only

first and second order

derivatives are

_needed,

and this is less tedious than other

_{approximations}

_suggested

by

eg, Mosteller and Wallace

(1964)

and

Lindley

(1980),

requiring

evaluation of

third derivatives. The

_posterior

variance can also be

approximated

by

finding

the

posterior

mean of

g

2

.

The

_only

modification needed is to define L* as

The

_{multiparameter}

case

When r is a _vector, as in this paper,

[48]

generalizes

to:

where

_r!

and

I’m

maximize L* and

_L,

_{respectively,}

and H* and H are minus the

inverse matrices of second derivatives of L* and L with respect to

_r,

evaluated at

r!

and

I’

m

,

respectively.

Marginal posterior

densities

The method can also be used to

_{approximate marginal posterior}

densities of individual _parameters of T. Partition r’ as

!T1, r

,2

j.

If the order of r is p, say,

then

T

2

is of order p -

1 (4

in our

case).

The

marginal posterior density

of

T

l

is:

where

₁

_{r(r l, r}

₂

₎

is the

_{joint posterior density}

of r. From

_preceding

_{developments,}

where

_I

_’m

_is

the mode of the

_posterior

distribution of

_T,

and

_{L&dquo;(r .. )}

is the matrix

of second derivatives of L with _respect to 7r. Then:

where

_l(r

_m

₎

is the

_{log-likelihood}

evaluated at

r

m

.

Hence,

[53]

becomes:

Consider now the numerator of

_(51!,

and write it as:

is a function where

T

1

is fixed. Define

)

l

(r

2m

r

to be the

_{(p - 1)}

x 1 vector that

maximizes this function. This maximizer can be found

_employing

the derivatives in the

_{appendix. Then,}

similar to

_(53!,

we can write.

where

_B&dquo;(r

_l

_,

_r

_2m

_(r¡))

is the

_{(p &mdash;}

₁₎

x

_{(p -}

₁₎

matrix of second derivatives of B

with _respect to _r2.

Taking

the ratio between

_[56]

and

_[54]

the

_posterior

_density

of

_T

₁

in

_[51]

is

approximately:

The moments of the

_posterior

distribution of

_I’i,

must be found

_numerically

.

Remarks

It has been shown that the method of

_Tierney

and Kadane

₍₁₉₈₆₎

has less error than the usual normal

_{approximation}

centered at the

_posterior

mode with the order of

_{approximation}

_being

_O(n-

₂

_).

_However,

it also

_requires

that the functions to be

expanded

be either unimodal or dominated

_by

a

_single

_mode,

so

_sample

size must

be

_{sufficiently large}

for this to hold.

The

_requirement

that

_g(T)

be a

_positive

function is restrictive.

_Tierney

and Kadane

₍₁₉₈₆₎

_pointed

out that for the

_{approximation}

to be accurate for a function

g

taking

both

positive

and

negative values,

the

posterior

distribution of g must

be concentrated almost

_entirely

on one side of the

_origin.

However,

Tierney

et al

(1988)

extended the method to

_apply

to

_expectations

and variances of

_non-positive

functions. To obtain a second-order

_{approximation}

to

_E[g(r)],

_they

used the method of

_Tierney

and Kadane

₍₁₉₈₆₎

to

_approximate

the moment

_generating

function

E{exp [!(F)]},

whose

_integrand

is

_positive,

and then the result was differentiated. Another

_difficulty

arises in the

_{approximation}

_[49]

to the

_posterior

variance of

_g(r).

Unless

_computations

are made with sufficient

_precision,

[49]

can have a

large

error or _{turn up}

_negative.

Similar

_problems

can arise in the

_computations

of

posterior covariance,

ie

as a covariance matrix

_computed

from

[58]

_may not be

_positive

semi-definite.

CONCLUSION

This paper presents

theory

and

_techniques

for

_carrying

out a

Bayesian analysis

of

dispersion

parameters in a univariate model for maternal effects.

_Hower,

implemen-tation of the methods

_suggested

here poses difficulties to

_{quantitative geneticists}

interested in

_analysis

of

_large

data sets. The

_development

of feasible

_computing

techniques

is a

challenge

to researchers in the area of

_application

of numerical

methods to animal

_breeding.

Research is

_underway

to

_identify

more

_{promising algorithms}

to

_approximate

marginal

moments of

_{posterior distributions,}

a non-trivial

problem

as new

tech-niques

are

_developed

and there is little indication on the choice to make for

es-timating

(co)variance

components under

_{&dquo;non-exchangeability&dquo;}

of model

_[1].

Re-cently

Gelfand and Smith

₍₁₉₉₀₎

and Gelfand et al

₍₁₉₉₀₎

described the Gibbs

sampler,

a

_potential

_competitor

of the methods

_presented

here.

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APPENDIX

First and second derivatives of the

_{log-posterior}

of all

_(co)variance

components

The

_log

of

_(11!

is:

Let M’ =

_{(0 ( I}

_2a

_0]

be a 2a x

_(p

+ 2a +

_d)

matrix such that

M’i

=

a. In the same

way, N’ =

[0

[ 0 [ Id]

be a d x

(p

+ 2a +

_d)

matrix such that

N’i

=

The

0

represents a matrix of

_appropriate

order with all elements

_equal

to zero.

To

_simplify

the

_derivation,

we will

_decompose

(A.1!

into _components, take deriva-tives with _respect to an element of

_G(g

_ij

say), (Tkm

or

!Eo,

and collect results to

obtain the desired

_expressions.

!,

Derivatives of

_{(y’y -}

0 W’y)

The term

_y’y

does not

_depend

on r. The other term is

8’W’t/

=

y’WCW’y,

_so that:

where

_E

_ij

is a 2 x 2 matrix with all elements

_equal

to _zero, with the

_exception

of a one in

_{position i,j.}

Note that if ei

)

j

(e

is a 2 x 1 vector with a 1 in the i-th

(j-th)

position

E2!

=

eie!.

The notation

_D(M

_l

_{, ... , M,l}

stands for a block

diagonal

matrix with the s blocks

_{being equal}

to

M

i

,

(i

=

_1,...,

s).

Since

i

=

CW’y

and

In a similar way

Second derivatives are obtained from

[A.3]

to

[A.5].

Additional second derivatives are:

Derivatives of

_log

_W’W

+

_£

_I

We use the result in Searle

(1979):

Using

[A.12],

the derivative of

_log

_W’W

+ E ! with

respect to _gij is

Other derivatives

We now consider the

_remaining

derivatives and these are:

with

_[A.12]

used to obtain the second term on the

right

of

[A.22].

Likewise

Second derivatives obtained from

_[A.22!-[A24]

are: