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Conditional Specification with Exponential PowerDistributionsPaloma Main a & Hilario Navarro ba Departmento de Estadística e I.O. , Universidad Complutense de Madrid , Madrid, Spainb Departmento de Estadística, I.O. y Calc. Num., UNED , Madrid, SpainPublished online: 10 Jun 2010.

To cite this article: Paloma Main & Hilario Navarro (2010) Conditional Specification with Exponential Power Distributions,Communications in Statistics - Theory and Methods, 39:12, 2231-2240, DOI: 10.1080/03610920903009392

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Communications in Statistics—Theory and Methods, 39: 2231–2240, 2010Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610920903009392

Conditional Specification with ExponentialPower Distributions

PALOMA MAIN1 AND HILARIO NAVARRO2

1Departmento de Estadística e I.O.,Universidad Complutense de Madrid, Madrid, Spain2Departmento de Estadística, I.O. y Calc. Num., UNED,Madrid, Spain

The problem of modeling Bayesian networks with continuous nodes deals withdiscrete approximations and conditional linear Gaussian models. In this article wehave considered the possibility of using the exponential power family as conditionalprobability densities. It will be shown that for some platikurtic conditionaldistributions in this family, conditional regression functions are constant. Theseresults give conditions to avoid compatibility problems when distributions withlighter tails than the normal are used in the description of conditional densities tospecify joint densities, like in Bayesian networks.

Keywords Bayesian networks; Conditionally specified distributions; Exponentialpower distributions.

Mathematics Subject Classification Primary 62E10, 62E15; Secondary 62H05.

1. Introduction

Bayesian network is a model that describes the joint distribution of randomvariables exploiting the dependence structure of the variables. Therefore, there is aqualitative part that encodes the dependence or independence of the variables, whichis represented by a directed acyclic graph (DAG) and a quantitative part that givesthe univariate conditional and marginal distributions expressed by factorizing thejoint distribution as follows:

f�x1� � � � � xn� =n∏

i=1

f�xi �pa�xi���

with pa�xi� as the set of parents of the variable Xi in the graph, pa�xi� ⊂�X1� � � � � Xi−1�. The characterization of a joint distribution by its univariate

Received August 26, 2008; Accepted April 30, 2009Address correspondence to Paloma Main, Departmento de Estadística e I.O.,

Universidad Complutense de Madrid, Pza. Ciencias, 3, Madrid 28040, Spain; E-mail:pmain@mat.ucm.es

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2232 Main and Navarro

conditionals is essential in this context in order to describe the elements of theproblem and also to perform inference in Bayesian networks. A general introductionto Bayesian networks is Jensen and Nielsen (2007). More technical treatment ofthe concepts and algorithms from graph theory used for Bayesian networks can befound in Castillo, Gutiérrez, and Hadi (1997) and Cowell et al. (2007). Some otheralgorithms were introduced by Pearl (1988) and Lauritzen and Spiegelhalter (1988).

In this setting, compatibility of conditional distributions to give a jointdistribution for some different families could be helpful to handle the quantitativepart of a Bayesian network. Looking at continuous cases, the multivariatenormal distribution was considered to define Gaussian Bayesian networks withthe corresponding normal univariate conditional distributions, linear regressionfunctions, and constant conditional variances. It is also known (Arnold, Castillo,and Sarabia, 1999) that there are other normal conditional distributions that arecompatible but these do not give normal joint densities. So, nonlinear regressionfunctions and nonconstant conditional variances will be needed. These models canhave multimodal densities (Arnold et al., 2000).

We have studied this problem for exponential power distributions as anextension of the normal model in order to include some slightly different behavioraround the mode. Note that for symmetric densities the degree of peakednessaround the mode involves different tail behavior. This is a family that dependson a kurtosis parameter that varies from leptokurtic to platikurtic distributionsconsidering the normal as a mesokurtic distribution. It was mentioned in Box andTiao (1973) for robustness studies in Bayesian inference and it was used later withthis purpose in many situations. Some other aspects are studied in DiCiccio andMonti (2004), Kuwana and Kariya (1991), and West (1987). Also, the normalppackage for the statistical environment R was developed in Mineo and Ruggieri(2005).

A multivariate generalization was introduced in Gómez, Gómez-Villegas, andMarín (1998) and some applications in Gómez, Gómez-Villegas, and Marín (2002),and Marín (2000). This multivariate family is used to extend the joint normalspecification for Bayesian networks in Main and Navarro (2009). However, we showthat the extension to non-Gaussian conditionals in continuous Bayesian networksmay carry noncompatibility problems.

The density function we are going to use for the univariate conditionals is

f�x � �� �� = k���

�exp{−12

∣∣∣∣x −

�

∣∣∣∣2�}

� −� < x < ��

for ∈ �� � > 0� � > 0, the location, scale, and kurtosis, parameters, respectively,and the constant k��� = �

� 12� �2

12�. Some particular cases are the normal, with �= 1,

the “peaked” double exponential, with � = 12 and the “flat” uniform as the limit

distribution for � → �.The problem of finding conditionally specified joint densities for normal

distributions comes from Bhattacharyya (1943) and was later studied with someother families of distributions. It deals with the possibility of characterizingmultivariate distributions, which are inherently difficult to visualize, by somefeatures of their conditional distributions. Essential general surveys with a completebibliography are Arnold, Castillo, and Sarabia (1999, 2001). In this context

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Conditional Specification 2233

we have studied the problem of conditional specification with exponential powerdistributions. Our main reason to analyze this problem is related to the possibilityof defining some other models for Bayesian networks with continuous nodes.However, our results led to the conclusion that conditional regression functionsshould be constant if we take the family of exponential power distributions forthe conditionals in Bayesian networks. The article is organized as follows: Sec. 2develops the problem of conditional specification in bivariate distributions; here themain result is shown to get the joint density corresponding to exponential powerconditionals with some particular parameters. In Sec. 3 a multivariate distributionis proposed. Finally, Sec. 4 presents some conclusions.

2. Bivariate Distributions

Let us suppose �X� Y� is a random variable that has a joint density f�x� y� withrespect to some product measure on ��, with marginal densities fX�x�, fY �y�and conditional densities fY � x�y�, fX � y�x�. The question is when is it true that therewill exist a joint density with exponential power conditionals such that

fY � x�y�−�<x<�

= k��1�

�1�x�exp{−12

∣∣∣∣y − 1�x�

�1�x�

∣∣∣∣2�1}

� −� < y < ��

fX � y�x�−�<y<�

= k��2�

�2�y�exp{−12

∣∣∣∣x − 2�y�

�2�y�

∣∣∣∣2�2}

� −� < x < �� (1)

for some 1�x� ∈ �� 2�y� ∈ �� �1�x� > 0� �2�y� > 0� �1 > 0, and �2 > 0. If thisdensity, f�x� y�, exists, then

fX�x�fY � x�y� = fY �y�fX � y�x� (2)

and it may be defined by either of above products.

2.1. Conditionally Specified Densities

The topic of conditional specification deals with the situation where the conditionaldensities fX � y�x� and fY � x�y� are specified and seek proper marginal densities fX�x�and fY �y� that satisfy (2). Necessary and sufficient conditions for compatibility, inthe continuous case, are provided by the following theorem (Arnold and Press,1989) where the families of candidate conditional densities are denoted by a�x� y� =fX � y�x� and b�x� y� = fY � x�y�.

Theorem 2.1. A joint density f�x� y�, with a�x� y� and b�x� y� as its conditionaldensities, will exist iff

(i) ��x� y� � a�x� y� > 0� = ��x� y� � b�x� y� > 0� = N and(ii) There exist functions u�x� and v�y� such that for every �x� y� ∈ N we have

a�x� y�

b�x� y�= u�x�v�y�

in which u�x� is integrable.

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2234 Main and Navarro

Then, for conditionals that are compatible, the appropriate marginal densitiesfX�x� and fY �y� must exist and be proportional to u�x� and inverse proportional tov�y�, respectively, with the integrability condition to make sure that the joint densityis proper.

Assuming constant conditional regression functions, the characterizationtheorem in Arnold and Strauss (1991) gives the next models for any real positive �1

and �2

f�x� y� ∝ exp−12

{��x − ��2�2 + ��y − ��2�1 + ��x − ��2�2 �y − ��2�1

}� x ∈ �� y ∈ ��

(3)

fX�x� ∝1

��+ ��x − ��2�2� 12�1

exp−12

{��x − ��2�2

}� x ∈ ��

(4)fY �y� ∝

1

��+ ��y − ��2�1� 12�2

exp−12

{��y − ��2�1

}� y ∈ �

with

1�x� = �� 2�y� = ��

�1�x� =(�+ ��x − ��2�2

)− 12�1� �2�y� = ��+ ��y − ��2�1�− 1

2�2 �

k��1� =�1

(

12�1

)2

12�1

� k��2� =�2

(

12�2

)2

12�2

�

for some � ∈ �� � ∈ �� � > 0� � > 0 and � > 0.Therefore, the conditionals are as (1)

fY � x�y�−�<x<�

= �1��+ ��x − ��2�2� 12�1

(

12�1

)2

12�1

exp{− 1

2��+ ��x − ��2�2��y − ��2�1

}� y ∈ ��

(5)

fX � y�x�−�<y<�

= �2

(�+ ��y − ��2�1� 1

2�2

(

12�2

)2

12�2

exp{−12��+ ��y − ��2�1��x − ��2�2

}� x ∈ ��

Also, it can be shown that

∫ ∫f�x� y�dx dy = 2

12�1 �

12�2

− 12�1

�1�2�1

2�2

(12�2

)

(12�1

)U

(12�2

� 1+ 12�2

− 12�1

���

2�

)

provided 1+ 12�1

− 12�2

� �−, with U�a� b� x� being the confluent hypergeometricfunction, defined in Abramowitz and Stegun (1968).

For the particular case �1 = �2 = 1 it yields the joint density obtained withconditional normal distributions in Arnold, Castillo, and Sarabia (1999) for thefollowing parameters:

B = �−����+ ��2�� C = ��+ c�2�� D = ��+ ��2��

E = −�v� F = v� G = �−����+ ��2�� H = 2�v�� J = −v��

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Conditional Specification 2235

related by

J

F= H

2E= G

D= −��

E

F= B

C= −��

This means five parameters instead of eight parameters that correspond to thegeneral model in Arnold, Castillo, and Sarabia (1999). This reduction is due to therestriction of constant conditional regression functions.

Also, allowing nonconstant regressions but assuming that 2�1 = m and 2�2 = n

are even integers larger than 2, it can be shown that the compatibility conditionleads to the conclusion that conditional regression functions must be constant.

Theorem 2.2. Given the conditional distributions as in (1), with m = 2�1 > 2,n= 2�2 > 2, and m�n even integers, the unique corresponding joint and marginaldensities are (3) and (4), respectively.

Proof. We are going to describe the general strategy; for more details see Main andNavarro (2007).

If we take logarithms in (2) for the models (1) with m�n even integers, it gives

logfX�x�k�m�

�1�x�− 1

2

(y − 1�x�

�1�x�

)m

= logfY �y�k�n�

�2�y�− 1

2

(x − 2�y�

�2�y�

)n

then

�2�1�x�mu�x�− �y − 1�x��

m��2�y�n = �1�x�

m�2�2�y�nv�y�− �x − 2�y��

n� (6)

with

u�x� = logfX�x�k�m�

�1�x� v�y� = log

fY �y�k�n�

�2�y�

and the problem comes when we look for the solutions,u�x�� v�y�� 1�x�� �1�x�� 2�y�, and �2�y�, if they exist. We can expand thebinomials of both sides and rearrange the results to get the final equation.Using the theorem of Aczel (1966) with sets of linearly independent functions��1�x�

m� x�1�x�m� � � � � xn�1�x�

m� and ��2�y�n� y�2�y�

n� � � � � ym�2�y�n� we get, as in

Arnold and Strauss (1991), the next results for the functional Eqs. (6)

�−1�x��m − 2�1�x�

mu�x�−� m

m−1 �1�x�m−1

���−� m

1 �1�x�1

= A�m+1��n+1�

�1�x�m

x�1�x�m

���xn�1�x�

m

(7)

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2236 Main and Navarro

�−2�y��n − 2�2�y�

nv�y�−� n

n−1 �2�y�n−1

���−� n

1 �2�y�1

= B�n+1��m+1�

�2�y�n

y�2�y�n

���ym�2�y�

n

(8)

with A = �aij� and B = �bij� matrices of coefficients satisfying A′ = B. Therefore,with the notation F ′

i � i = 1� � � � � m+ 1 and G′j� j = 1� � � � � n+ 1 for the rows of A

and B, respectively, we have

(F1� F2� � � � � Fm� Fm+1

) = G′

1· · ·G′

n+1

� (9)

Now, we solve (7) to get the solutions. Then, denoting x∗′ = �1� x� � � � � xn�, firstlywe have

�1�x� =1

�F ′m+1x

∗�1m

= 1

�am+1�1 + am+1�2x + · · · + am+1�n+1xn�

1m

�

and then,

1�x� = − F ′mx

∗

mF ′m+1x

∗ � (10)

For the current problem, m > 2, there are some other conditions to be imposed thatdiffer from the normal case �m = 2�; these are

(m2

)1�x�

2 = F ′m−1x

∗

F ′m+1x

∗ � � � � �(mj

)�−1�x��

j

= F ′m−j+1x

∗

F ′m+1x

∗ � � � � �

(m

m− 1

)�−1�x��

m−1 = F ′2x

∗

F ′m+1x

∗

and all of them have to be equal to the corresponding jth power of (10), so that theconditions are

(F ′mx

∗

F ′m+1x

∗

)j

= mj

� mj �

F ′m−j+1x

∗

F ′m+1x

∗ � j = 1� � � � � m− 1

then

mj

� mj �

F ′m−j+1x

∗(F ′m+1x

∗)j−1 = (F ′mx

∗)j� j = 1� � � � � m− 1�

For j = 2 the previous condition gives x∗′ 2mm−1Fm−1F

′m+1x

∗ = x∗′FmF′mx

∗ for everyx∈�; then the matrices of both sides have to be equal, but 2m

m−1Fm−1F′m+1 = FmF

′m

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Conditional Specification 2237

implies the proportionality of the vectors considered; that is, for some � ∈ � wehave Fm = � 2m

m−1Fm+1 and Fm−1 = �2 2mm−1Fm+1. In general, we find that

Fm−j+1 =(mj

)(2�

m− 1

)j

Fm+1� j = 1� � � � � m− 1�

Thus, the matrix of coefficients A has proportional rows and

1�x� = − F ′mx

∗

mF ′m+1x

∗ = − 2�m− 1

so that the regression function has to be constant.Also, by (9) B has proportional columns; denoting y∗′ = �1� y� · · · � ym� and

solving (8) we get again proportional rows in B. Then, similar results as above forA are obtained for B with

2�y� = − 2�∗

n− 1

Summing up, the matrices of coefficients only depend on

�� �∗� a11� a1�n+1� am+1�1� am+1�n+1

and the solutions are as follows

�1�x� =1{−am+1�1 − am+1�n+1

[(x + 2�∗

n−1

)n − ( 2�∗n−1

)n]} 1m

�

u�x� = 12

{[a11 −

(2�

m− 1

)m

am+1�1

]+

+[a1�n+1 −

(2�

m− 1

)m

am+1�n+1

] [(x + 2�∗

n− 1

)n

−(

2�∗

n− 1

)n]}�

�2 �y� =1(−a1�n+1 − am+1�n+1

[�y + 2�

m−1 �m − ( 2�

m−1

)m]) 1n

�

v�y� = 12

{[a1�1 − a1�n+1

(2�∗

n− 1

)n]

+[am+1�1 − am+1�n+1

(2�∗

n− 1

)n] [(y + 2�

m− 1

)m

−(

2�m− 1

)m]}�

From the conditions �1�x� > 0 and �2�x� > 0, the constants must satisfy

am+1�1 ≤ am+1�n+1

(2�∗

n− 1

)n

≤ 0�

a1�n+1 ≤ am+1�n+1

(2�

m− 1

)m

≤ 0�

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2238 Main and Navarro

Then, the unique models compatible with (2) are given by (3), being � = − 2�m−1 , � =− 2�∗

n−1 , � = 1{−[a1�n+1−am+1�n+1

(2�

m−1

)m]} 1n, � = −am+1�n+1, � = −am+1�1 + am+1�n+1

(2�∗n−1

)n,

with some restrictions over the constants to make fX�x� and fY �y� density functions.They can be also obtained from Arnold and Strauss (1991) with the condition ofconstant conditional regression functions for m�n > 0. �

Then, we must have exponential power conditional distributions as in (5)to make them compatible. Thus, the variables are uncorrelated with boundedconditional variances

V �Y �X = x� = 22m � 3

m � � 1

m ���+ � �x − ��n�−

2m�

V �X � Y = y� = 22n � 3

n � � 1

n ���+ ��y − ��m�−

2n�

Note also that � = 0 implies independence.

3. Multivariate Distributions

If we have to impose, as shown above, constant conditional regression functions,then we are dealing with conditionals in exponential families. Therefore, themultivariate extension of the model is (Arnold, Castillo, and Sarabia, 1999)

f�x1� � � � � xp� ∝ exp1∑

i1=0

1∑i2=0

· · ·1∑

ip=0

�i1i2���ip

p∏j=1

��xj − j�nj �ij �

xi ∈ �� i = 1� � � � � p� (11)

with i, location parameters, and coefficients �i1i2���ip to be chosen to ensure thatall the conditional variances are always positive and the density is integrable.Consequently, the full conditionals have to be, for i = 1� � � � � p and xi ∈ ��

f �xi � x−i� ∝ exp−12

∑�ij=0�1j �=i�

�i1�i2����1����ip

p∏j=1j �=i

(∣∣xj − j

∣∣nj )ij �xi − i�ni

that is, exponential power distributions with

i�∑

�ij=0�1j �=i�

�i1�i2����1����ip

p∏j=1j �=i

(∣∣xj − j

∣∣nj )ij � ni

as location, scale, and kurtosis parameters, respectively. If we want to useconditionals in a family that enlarge the normal distributions, the conditionalsabove could also be used to model Bayesian networks with a joint density belongingto the family (11).

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Conditional Specification 2239

4. Summary

A family of conditional distributions that includes the normal can be consideredto obtain a conditionally specified joint density. This is the family of exponentialpower distributions that extends the normal model including lighter and heaviertailed distributions. Some particular distributions of this family fulfill a theorem ofexistence and uniqueness of the joint density. The results led us to the conclusionthat the conditional regression functions must be constant. Then, a slightly differentmodel from normal distribution can be used for conditionals in a continuousBayesian network. However, constant conditional means and varying conditionalvariances should be restricted in order to achieve compatible models.

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