A useful reparameterisation to obtain samples from conditional inverse Wishart distributions

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A useful reparameterisation to obtain samples from

conditional inverse Wishart distributions

Inge Riis Korsgaard, Anders Holst Andersen, Daniel Sorensen

To cite this version:

Note

A

useful

_{reparameterisation}

to

obtain

samples

from conditional

inverse

Wishart

distributions

Inge

Riis

_Korsgaard

Anders Holst Andersen

Daniel

Sorensen

a

Department of Animal

_Breeding

and

_Genetics,

Research Centre

_Foulum,

PO Box _50, DK-8830

_Tjele,

Denmark

i’

Department

of Theoretical

_{Statistics, University}

of

_Aarhus,

DK-8000 Aarhus

C,

Denmark

(Received

12 December

1997; accepted

6

_January

₁₉₉₉₎

Abstract - A

_{Bayesian joint analysis}

of

_normally

distributed traits and

_binary

traits,

using

the Gibbs

_sampler,

requires the

_drawing

of

_samples

from a conditional inverse

Wishart distribution. This is the

_fully

conditional _posterior distribution of the residual covariance matrix of the

_normally

distributed traits and liabilities of the

binary

traits.

_{Obtaining samples}

from the conditional inverse Wishart distribution is not

straightforward.

However,

combining

well-known matrix results and

_properties

of the

Wishart distribution, it is shown that this can be

_easily

carried out

_{by successively}

drawing

from Wishart and

_normally

distributed random variables.

_©

_{Inra/Elsevier,}

Paris

conditional inverse Wishart distribution

_/

Gibbs sampling

/

binary traits /

residual covariance matrix

Résumé - Reparamétrisation permettant d’obtenir des échantillons tirés d’une

loi de Wishart inverse conditionnée. Une

analyse bayésienne

utilisant l’échantillon-nage de Gibbs, de caractères distribués normalement conjointement avec des

carac-tères

_{binaires, requiert}

le

_tirage

d’échantillons dans une loi de Wishart inverse

conditionnée. Il

_s’agit

de la distribution a

_posteriori

de la matrice de covariance

résiduelle des caractères distribués normalement et des variables latentes

corres-pondant

aux variables binaires. L’obtention d’échantillons

correspondants

n’est _pas

évidente.

Cependant

l’utilisation de résultats bien connus sur les matrices et des

propriétés

de la distribution de Wishart permet d’aboutir à une solution en tirant *

Correspondence

and

_reprints

successivement dans une loi de Wishart et dans des lois

_{gaussiennes. ©}

_{Inra/Elsevier,}

Paris

distribution de Wishart inverse conditionnée / échantillonnage de Gibbs

_/

caractères binaires

/

matrice de covariance résiduelle

1. INTRODUCTION

Markov chain Monte Carlo makes

possible

the

_exploration

of

_posterior

distributions with relative _ease,

_using

models which are

computationally

too

complex

to be

_implemented

with other

_approaches.

A case in

_point

is the models for a

_joint

_analysis

of a

_normally

distributed trait

(such

as

_weight

_gain

or

yield

of

_milk)

and a

binary

trait

(resistant

or not resistant to

_disease,

twin or

_single

birth in

_cattle)

-- where the

_binary

trait is modelled via the threshold model

_!9!,

which invokes the existence of an unknown

continuously

distributed

underlying

variable,

the

_liability.

A

_{Bayesian analysis}

of such

_traits,

_using

the Gibbs

sampler, requires

the

_drawing

of

_samples

from a conditional inverse Wishart

distribution

_(e.g.

_{!3, 5, 8!).}

This is the

_fully

conditional

_posterior

distribution of the residual covariance matrix of the

_normally

distributed traits and liabilities of the

binary

traits.

_{Obtaining samples}

from the conditional inverse Wishart distribution is not

_{straightforward.}

The _purpose of this note is to

_present

an easy method to obtain

_samples

from the conditional inverse Wishart

distribution,

where the

conditioning

is on a block

_diagonal

_submatrix,

R

22

,

equal

to the

_identity

matrix of the inverse Wishart distributed

_matrix,

R =

Ri

l

R1

2

This is carried out

Bit

21

21

R

22

/

by combining

well-known

_{relationships}

between a

_partitioned

matrix and its

inverse and

_properties

of Wishart distributions. The

_proposed

method can

alternatively

be arrived at

_{by using}

both another

_{reparameterisation}

and the

properties

of the inverse Wishart distribution. This was carried out in Dr6ze and Richard

_[2]

and is well-known in the econometric literature. The need for

sampling

from a conditional inverse Wishart distribution is motivated

by

a

Bayesian

multivariate

_analysis

of pi

normally

distributed traits and p2

binary

traits,

pl, p2>

1, using

the Gibbs

sampler

and data

_{augmentation.}

2. THE MODEL

Assume that _PI

_normally

distributed traits and _p2 traits with

_binary

response are observed for each animal. Data on animal i are _yi =

(Y;1’Y;2)&dquo;

where y21! is the observed value of the

_{jth normally}

distributed

trait, j =

1, ... , pl,

and _{!2zk is} the observed value of the kth

_{binary trait,}

=

_{l, ... , !2. It}

is assumed that the outcome of

_Y

_zk

is determined

by

an

_underlying

continuous

random

variable,

the

_liability,

_U

_ik

_,

where

Y;

2k

= 1 if

U

ik

_>Tand

Y;

2k

= 0 if

U

Zk

< T, where T is a fixed

threshold,

often assumed to be

_equal

to zero. Let

Wi

=

(Y;

l’

Ui)’

and define W as the

np-dimensional

column vector,

containing

the

_W

_i

_s,

W’ =

_(W!,...,

_W

_!

_{y ),}

_p =

PI

where X and Z are

_design

matrices

_associating

W with ’fixed’

effects, b,

and additive

_{genetic values,}

_a,

_{respectively.}

The usual

_condition,

_(R2z)!!

= 1

(e.g.

[1]),

has been

_imposed

in the conditional

_probit

model for

_Y

_i2k

_given

_a, k =

_{1, ... ,}

p2. Furthermore it is assumed that liabilities of the

binary

traits are

conditionally independent, given

b and a. The

following

_prior

distributions are assumed: b is

_uniform,

and that

_RIR

₂₂

=

Ip

2

follows a conditional inverse Wishart distribution with

density

up to

_{proportionality given by:}

Augmenting

with the vector U =

(U

1

, ... , U

n

)’

of

liabilities,

and also

as-suming

that a

_priori

_b,

(a, G)

and R are

_{mutually independent,}

it follows

(e.g.

[5]),

that the

_fully

conditional

_posterior

distributions

_required

to

im-plement

the Gibbs

sampler

are _easy to

_sample

from with the

_exception

of the

_fully

conditional

_posterior

distribution of

_RIR

₂₂

=

Ip

2

.

The

_fully

condi-tional

_posterior

distribution of

_RIR

₂₂

=

Ip

2

is conditional inverse Wishart distributed with

_density

_proportional

to

_equation

₍₂₎

with

_E

_R

_replaced

_by

/ 7t _B -1

freedom

_f

_R

_{replaced by f}

_R

_,

+ n. In the method to be

_proposed

for

_sampling

from

_equation

₍₂₎

in a

_{computationally simple}

_manner, the

_properties

sum-marised below are essential.

Assume that

_R-7!(E,/)

and let

_{V = R-}

₁

_,

then

_{V - W, (E, f ).}

Further-more, define T =

, T

)

3

z

, T

1

(T

by

T

l

=

V

ll

,

T

2

=

V

11

1

V

1

z,

and

_T

₃

=

V

22

.

1

= !22 -V2iVii!Vi2;

where V

= (

V

21

Vlz

J

is

a

_{partitioning of V; V}

_ll

is

V21

21

V22

pi x _pi and

_V

₂₂

is P2 x _pz. Then the

_following

results hold:

Result 1: there is a one to one

_relationship

between T and R

_{given by}

Result 5:

_(T

_l

_{, T}

₂

₎

is

_independent

of

_T

₃

_,

which

_implies

that the conditional distribution of

_(T

₁

_{, T}

_Z

₎

_given

_T

₃

= _t3 is

_equal

_to the

_marginal

distribution of

(T

I

, T

2

)

Result 1 is immediate. Results

_2,

4 and 5 all can be found in Mardia et al.

3. AN ALTERNATIVE PARAMETERISATION

Let R -

_IW

_P

_{(E, f}

_R

+

_n)

be

_{reparameterised}

in terms of

_(T

₁

_,

_T

₂

_,

_T

₃

₎

_given

by

result 1:

with the distribution of

_(T

_i

_,

_T

₂

_,

_T

₃

₎

as

_specified

in results

_{2, 3,}

4 and 5.

The distribution of

_{R! (R22}

=

Ip

J

is that of

R! (T3

=

Ip

2

)

’

This follows because

T

3

=

R22

is

_{a one to one} transformation

of R

z2

(property (10.4.3)

from

calculus of conditional distributions in

_{Hoffmann-Jorgensen}

_(4!)

and because of result 6. Next

_inserting

_T

₃

=

I

P2

in R

_(property

_(10.4.4)

in

_{Hoffmann-Jorgensen}

!4!)

it follows from result 5 that the distribution of

_R!(T3

=

Ip

2

)

is that of

From above it follows that if

t

1

is

_sampled

from

_{Ti -}

_W

_Pl

_(1:

₁₁

_{, f}

_R

_{+ n),}

next

₂

_t

from

_T

₂₁

_T

_l

=

_{t1 !}

/1:1

l

, t1

2

l

1

2 (1:

XP

NP

1 ! £

22

. i ) ,

then

is a realised matrix from the conditional inverse Wishart distribution of R

_given

R

22

=

_I

_p

_2’

4. CONCLUSION

We have

presented

a

simple

method to draw

samples

from conditional inverse Wishart distributions. The

_conditioning

is on

_R

₂₂

_equal

to the

_identity

matrix,

where R =

R2

1

R.12

)

is

_{a partitioning}

of an inverse Wishart

R21

R22

distributed matrix. The method is relevant in a

_{Bayesian joint analysis}

of

normally

distributed and

_binary

traits

_(the

latter with associated

_{liabilities),}

using

the Gibbs

sampler.

The

_methodology

was illustrated based on models

with additive

_genetic

effects

_only.

The

_{generalisation}

to several random effects is immediate.

ACKNOWLEDGEMENT

REFERENCES

[1]

Cox

_D.R.,

Snell

_{E.J., Analysis}

of

_Binary

Data,

Chapman

and

_{Hall, London,}

1989.

[2]

Drèze

J.H.,

Richard

_{J.-F., Bayesian analysis}

of simultaneous

_equation

_systems,

in: Griliches

_{Z., Intriligator}

M.D.

_(Eds.),

Handbook of

Econometrics,

North-Holland

Publishing Company,

vol. 1, 1983, pp. 587-588.

[3]

Jensen

_{J., Bayesian analysis}

of bivariate mixed models with one continuous and one

_binary

trait

_using

the Gibbs

sampler, Proceedings

of the 5th World

Congress

on Genetics

_Applied

to Livestock Production 18

₍₁₉₉₄₎

333-336.

[4]

Hoffmann-Jorgensen

J.,

Probability

with a View toward

Statistics, Chapman

and

_Hall,

New

_York,

1994.

[5]

Korsgaard LR.,

Genetic

analysis

of survival

_data,

Ph.D.

_{thesis, University}

of

Aarhus, Denmark,

1997.

[6]

Lauritzen

_{S.L., Graphical Models,}

Oxford

_{University Press,}

New

York,

1996.

[7]

Mardia

_K.V.,

Kent

_{J.T., Bibby J.M.,}

Multivariate

_Analysis,

Academic

_Press,

Great

_Britain,

1979.

[8]

Sorensen

_D.,

Gibbs

_sampling

in _quantitative

_genetics,

Internal _report no. 82

from the Danish Institute of Animal

_Science,

1996.

[9]

Wright S.,

An

_analysis

of

_variability

in number of

_digits

in an inbred strain of