Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Objective Bayes Factors for Informed Hypotheses:
”Completing” The Informed Hypothesis and
”Splitting” the Bayes Factors
David Torres N´nez12 Luis Ra´l Pericchi Guerra12 Guimei Liu 1
u˜ u
Department of Mathematics, University of Puerto Rico at Rio Piedras Campus,
P.O. Box 23355, San Juan PR 00931-3355. lrpericchi@uprrp.edu
BBC Biostatistics and Bioinformatics Core Cancer Center of the UPR
March 2009
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Contents
1 Introduction
2 General Formalism or Coherent Approach
Bayes Factor and Posterior Model Probabilities
Bayes Factors
Posterior Model Probabilities
3 Informative Hypotheses
4 Illustration: Dissociative Identity Disorder
DID Zellner-Siow Conventional Prior
5 Conclusions
6 Acknowledgements
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Introduction
Informed Hypotheses are particular hypotheses about the vector of
means µ in the Classical Normal ANOVA Model:
Y = Xµ + ,
where is the vector of uncorrelated Normal errors with constant
variance σ 2 .
Informed Hypotheses involve a complex ordering of means like, say:
H1a : µa = µb < µc < µd = µe ,
or,
H1b : µa < µb = µc < µd < µe ,
among other possibilities.
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
In order to compare models, in the canonical Bayesian way (using
Bayes Factors), and upon specification of a prior under say, H1a ,
π(µa , µb , µc , µd , µe , σ|H1a ) we need to calculate the marginal
densities:
m(y|H1a ) = f (y|µ, σ, H1a )π(µa , µb , µc , µd , µe , σ|H1a )dµdσ.
H1a
(2.1)
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
There are two difficulties with equation 2.1:
1 The difficulty of assessment of the prior π(.|H1a ) which is
potentially highly influential.
2 The difficulty in the computation of integral in 2.1, due to the
awkward region of integration which is in the very definition
of an informed hypothesis.
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
A direct approach to deal with inequality constraints is in
[Berger and Mortera 1999] who study in depth the simplest
examples with inequality constraints. However their detailed
and incisive direct approach seems bound to be restricted to
very simple situations, unsuited to the kind of examples
encountered for example in Psychology, particularly for
dimensions greater than 2.
Another general approach is in (Klugkist 2008), who suggest
an Encompassing Approach plus the replacement of equality
constraints by constraints that the means are close.
Our own approach, that we call ”Completing and Splitting”
the informed hypothesis.
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Take for instance H1a with a ”completed” hypothesis
∗
H1a : µa = µb = µc = µd = µe ,
∗
with a corresponding ”completion” of H1b denoted by H1b , on
which all order constraints are replaced by inequalities. These
completions allows to split the problem in 2.1 in two factors: one
of an usual ”objective” marginal density, and (objective) Bayes
Factors and the other factor as a probability of a region with
positive measure. Thus the name of ”Completing and Splitting”
approach. In the sequel, it will be shown that:
∗ ∗
m(y|H1a ) = m(y|H1a ) × P r(H1a |y, H1a ), (2.2)
which is the pivotal identity of the ”completion and splitting”
approach. Equation 2.2 entails a substantive methodological
simplification to 2.1
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach Bayes Factor and Posterior Model Probabilities
Informative Hypotheses Bayes Factors
Illustration: Dissociative Identity Disorder Posterior Model Probabilities
Conclusions
Acknowledgements
Suppose that we are comparing q models for the data y,
Hi : Y has density fi (y|θi ), i = 1, . . . , q,
where the θi = (µi , σ 2 ) are the unknown model parameters.
Suppose that we have available prior distributions, πi (θi ),
i = 1, . . . , q, for the unknown parameters. Define the marginal or
predictive densities of Y,
mi (y) = fi (y|θi )πi (θi ) dθi .
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach Bayes Factor and Posterior Model Probabilities
Informative Hypotheses Bayes Factors
Illustration: Dissociative Identity Disorder Posterior Model Probabilities
Conclusions
Acknowledgements
A central quantity for comparing models: The Bayes factor of Hj
to Hi is given by
mj (y) fj (y|θj )πj (θj ) dθj
BFji = = .
mi (y) fi (y|θi )πi (θi ) dθi
The Bayes factor is often interpreted as the “evidence provided by
the data in favor or against model Hj vs the alternative model
Hi ”, but it should be remembered that Bayes Factors depend also
on the priors. In fact assuming an objective Bayesian viewpoint,
the priors may be thought of “weighting measures” and the Bayes
Factor as the “weighted averaged likelihood ratio”, which are in
principle more comparable than maximized likelihood ratios, as
measures of relative fit.
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach Bayes Factor and Posterior Model Probabilities
Informative Hypotheses Bayes Factors
Illustration: Dissociative Identity Disorder Posterior Model Probabilities
Conclusions
Acknowledgements
If prior probabilities P (Hi ), i = 1, . . . , q, of the models are
available, then one can compute the posterior probabilities of the
models from the Bayes factors. Using Bayes Rule, it is easy to see
that posterior probability of Hi , given the data y, is
 −1
q
P (Hi )mi (y) P (Hj )
P M P (Hi | y) = q = BFji  .
j=1 P (Hj )mj (y) P (Hi )
j=1
(3.1)
A particularly common choice of the prior model probabilities is
P (Hj ) = 1/q, so that each model has the same initial probability,
but there are other possible choices, see for example
[Berger and Pericchi 1996a].
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
We now proceed to apply the Objective Bayes Factor methods for
Informed Hypotheses. Suppose we have say 4 groups, with data
yi,j , i = 1, . . . , 4; j = 1, . . . , ni , and the data are assumed to be
Yi,j ∼ N (µi , σ 2 ).
The Null Hypothesis is:
H0 : µ1 = µ2 = µ3 = µ4 , σ > 0
against an informed hypothesis like:
H1a : µ3 < µ1 = µ4 = µ < µ2 , σ > 0.
This is not a standard comparison, and has generated considerable
interest under the name of Informed Hypotheses important among
other areas in Psychology.
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Alternatively, re-parametrizing H1a calling θ = µ, θ1 = µ − µ3 and
θ2 = µ2 − µ, the alternative hypothesis becomes:
H1a : θ1 > 0 and θ2 > 0, and the ”parameters under test” are
[θ1 , θ2 ]
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
We need to compute, calling θa all the mean parameters under
H1a :
N
N H1a f (y|θa , σ)π (θa , σ)dθa dσ
BFH1a,0 = . (4.1)
f0 (y|θ0 , σ)π N (θ0 , σ)dθ0 dσ
Notice that using π N (θa , σ) in (10), makes the comparison
without a proper scaling, so we also have to compute a Correction
Factor CF0,1a , so that
N
BFH1a,0 = BFH1a ,0 × CF0,H1a . (4.2)
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
”Completing and Splitting”
Informed Hypotheses.
1 ∗
Completing: First of all, it is useful to denote by H1a the
Hypothesis formed by replacing any inequality order
constraints by inequalities. For example, regarding the
illustration above, where H1a : µ3 < µ1 = µ4 = µ < µ2 then
∗
H1a : µ3 = µ1 = µ4 = µ = µ2 .
2 ”Splitting”: Denote by Θa the parameter space, which
∗
contains H1a .
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Lemma
It turns out that,
N N ∗ mN (y)
1a ∗
BFH1a,0 = BFH1a ,H0 ×P r(H1a |y, H1a ) =
∗
N (y)
×P r(H1a |y, H1a )
m0
(4.3)
where
∗ H1a f (y|θa , σ)π N (θa , σ)dθa dσ
P r(H1a |y, H1a ) = ,
∗
H1a f (y|θa , σ)π N (θa , σ)dθa dσ
mN (y) =
1a f (y|θa , σ)π N (θa , σ)dθa dσ,
∗
H1a
mN (y) =
0 f (y|θ0 , σ)π N (θ0 , σ)dθ0 dσ.
H0
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
The example on inter-identity amnesia in dissociative identity
disorder is taken from [Huntjens et al 2006].
Inter-identity amnesia in dissociative identity disorder: a simulated
memory impairment? in Psychological Medicine,36, 857-863.
The Hypotheses: First of all the Hypothesis of Equal Means:
H0 : µ1 = µ2 = µ3 = µ 4 .
Its negation is H2 : not H0 .
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Then,
H1a : µcontrol > {µtrue−amnesiacs , µDID−patients } > µDID−simulators
, that is
µ3 < µ4 = µ1 = µ < µ2 ,
or equivalently
H1a : µ3 < µ < µ2 .
Finally,
H1b : µcontrol > µtrue−amnesiacs > {µDID−patients , µDID−simulators }
µ1 = µ3 = µ < µ4 < µ2 ,
or
H1b : µ < µ4 < µ2 .
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Before going into the methods, in Table the Residual Sum of
Squares of the Null and ”Completed” hypotheses are presented:
Models Residual Sums Squares
H0 :RSS0 2196.4681
H2 :Not H0 : RSS2 237.62947
∗
H1a :RSS1a 260.47545
∗
H1b :RSS1b 253.83636
Table: Residual Sum of Squares of Completed Hypotheses
Note: RSS2: Any of the means different. For the Completed
∗ ∗
hypotheses we see that H1b is doing somewhat better than H1a .
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Transformation of the Design Matrix For the Zellner and
Siow Priors As in the Introduction we have the ANOVA model:
Y = Xβ + ,
on which, the first column of X should be a column of ones;
∗
Denote β = [β0 , α1 , α2 ]. The hypothesis H1a : µ3 = µ = µ2 should
become: α1 = 0; α2 = 0. In the Conventional Prior set up, a
proper conditional prior is given to the parameters under test (the
α’s), but an improper uniform prior is given to the common
parameter β0 .
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
For this to be justifiable as a Bayes Factor, then the first column of
the matrix X ought to be orthogonal to the other columns, i.e.
xt · xi = 0, i = 2, 3. We denote by ni the number of observations
1
in the group i. So we have n3 = 25 cases with mean µ3 , n2 = 25
cases with mean µ2 and n = 94 is the total number of cases, since
n1 = 19 and n2 = 25. For H1a , in the column of observations we
place first the observations of the first and fourth group (assumed
equal), secondly the observations of the third group (presumed to
be lowest) and finally the observations belonging to the second
group (presumed to be highest).
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Recall that the first and the next two columns of the design matrix
ought to be orthogonal. So the design matrix X1a for H1a is set to
be:  
1 −n3 /n −n2 /n
 . . .
 . . . 
 . . . 

 1
 −n3 /n −n2 /n  
 1
 (1 − n3 /n) −n2 /n  
X1a =  . . .
 . . . .

 . . . 
 1
 (1 − n3 /n) −n2 /n  
 1
 −n3 /n (1 − n2 /n) 

 . . .
 . . . 
. . . 
1 −n3 /n (1 − n2 /n)
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Notice that xt · [x2 , x3 ] = 0. It turns out, as can be checked after
1
simplifications, that:
H1a : µ3 < µ < µ2 is equivalent to: α1 < 0, α2 > 0.
Numerically (which will be needed for the bivariate Cauchy to be
used) it obtains that:
 
0.011 0 0
[X1a X1a ]−1 =  0
t
0.063 0.023  ,
0 0.023 0.063
and [X1a X1a ]−1 is the two by two lower right symmetric
∗t ∗
sub-matrix.
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Now we turn to the prior assessments: First the hypotheses are:
∗
H0 : α1 = α2 = 0; H1a : α1 and α2 = 0, H1a : α1 < 0, α2 > 0.
Clearly the design matrix under the Null Hypothesis is just a vector
of ones. It is customary here to assume references priors, under H0
∗
and conditionally proper under H1a :
N N
π0 (µ, σ) = 1/σ, and π1 (µ, α1 , α2 , σ)
= Cauchy[(α1 , α2 )|(0, 0), (X1a X1a /n)−1 σ 2 ] · 1/σ.
∗t ∗
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Computing prior probabilities with the Zellner and Siow Prior
for Bayes Factors, applied to Informed Hypotheses Zellner and
Siow Prior for model selection is, in our case of Informed
Hypotheses equal to:
det[X ∗t X ∗ /(nσ 2 )]
π(α|σ) = c , (5.1)
(1 + αt X ∗t X ∗ α/(nσ 2 ))3/2
where c = Γ(3/2)/π 3/2 and αt = (α1 , α2 ).
The distribution (26) is a bi-dimensional Cauchy with location zero
and scale (quasi variance) matrix equal to (X ∗t X ∗ /(nσ 2 ))−1 .
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Turning now for the other hypothesis:
H1b : µ1 = µ3 = µ < µ4 < µ2 ,
we place in the vector y of observations, first the amalgamated
first and third group, secondly the fourth group (assumed in the
middle) and finally the second group (presumed highest).
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
For H1b we set the following design matrix:
 
1 −(n2 + n4 )/n −n2 /n
 . . .
 . . . 
 . . . 

 1
 −(n2 + n4 )/n −n2 /n 

 1 (1 − (n2 + n4 )/n) −n2 /n 
 
 .. .
. .
.
X1b =  . .

 . . 
 1 (1 − (n2 + n4 )/n) −n2 /n 
 
 1 (1 − (n2 + n4 )/n) (1 − n2 /n) 
 
 . . .
 . . . 
. . . 
1 (1 − (n2 + n4 )/n) (1 − n2 /n)
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Notice that xt · [x2 , x3 ] = 0. It turns out, as can be checked after
1
simplifications, that:
H1b : µ1 = µ3 = µ < µ4 < µ2 is equivalent to: α1 > 0, α2 > 0.
Numerically (which will be needed for the bivariate Cauchy to be
used) it obtains that:
 
0.011 0 0
[X1b X1b ]−1 =  0
t
0.063 −0.04  ,
0 −0.04 0.08
and [X1b X1b ]−1 is the two by two lower right symmetric
∗t ∗
sub-matrix.
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
From the a posteriori densities and simulating via WINBUGS
(modeling a Cauchy density as a scale mixture of a Normal
and a Gamma, or using the ”zeroes trick”) we get the results for
H1a and H1b in the Tables 5 and 6.
H1a ZS prior H1a posterior H1a
p 0.19022 1.0
Table: Posterior and Prior Probabilities Implied by the Zellner and Siow
∗
prior for H1a given H1a
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
H1b ZS prior H1b posterior H1b
p 0.15349 1.0
Table: Posterior and Prior Probabilities Implied by the Zellner and Siow
∗
Prior for H1b given H1b
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses Zellner-Siow Conventional Prior
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
For the Bayes Factors in formula(19), there is the following
approximate formula in [Zellner and Siow 1980]
BF12 = π 1/2 /Γ((k1 + 1)/2)((n − k1 )/2)k1 /2 (R1 /R0 )(n−k1 −1)/2
Z
In Table 4 we display the Bayes Factors and Posterior Model
Probabilities for Zellenr and Siow’s prior.
Models BF Zellner-Siow Overall Posterior Prob.
H0 VS H2 2.83249e-040 P M P (H0 |y) =1.716e-039
H0 VS H1a 6.14075e-039 P M P (H1a |y) = 0.279
H0 VS H1b 2.38130e-039 P (H1b |y) = 0.721
Table: Posterior Probabilities given by Zellner and Siow Prior for
H0 , H1a , H1b
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
1 The Informed Hypothesis Problem becomes feasible when it is
split in two problems, first by ”completing” the informed
hypothesis and secondly by ”splitting” into two factors:
a Ratio of Objective Bayes Factor of appropriate Encompassing
Hypotheses that we call ”completions” of the Informed
Hypotheses, and
a Ratio of Prior to Posterior Probabilities of the Informed
Hypothesis.
2 The First Factor can be dealt with the usual methods of
Objective Bayes Factors. We have presented several
techniques, so that researchers can implement them as they
judge more appropriate.
3 The Second Factor can be expressed, with some ingenuity, as
an MCMC problem of estimation.
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
The authors were supported by National Science Foundation grants
DMS-0604896 and DUE-0630927. The authors are grateful to Dr.
M.E. P´rez, and Dr. John Cook for useful discussions and to the
e
Associate Editors.
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u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Abramowitz, M. and Stegun, I.: Handbook of Mathematical
Functions. National Bureau of Standards. Applied
Mathematics Series, 55, (1970)
Berger, J. and Mortera, J.: Default Bayes Factors for
One-Sided Hypothesis Testing. Journal of the American
Statistical Association, 31, 542–554 (1999)
Berger, J. and Pericchi, L.: The Intrinsic Bayes Factor for
Model Selection and Prediction. Journal of the American
Statistical Association, 91, 109–122 (1996)
Berger, J. and Pericchi, L.: The Intrinsic Bayes Factor for
Linear Models. In: J. M. Bernardo, et. al. (ed) Bayesian
Statistics 5. Oxford University Press, 23–42, London (1996)
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

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Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Berger, J. and Pericchi, L.: On the Justification of Default and
Intrinsic Bayes Factors In: J. C. Lee, et. al.(ed) Modeling and
Prediction. Springer, Berlin Heidelberg New York,276–293
(1997)
Berger J., and Pericchi, L.: Objective Bayesian Model
Methods for Model Selection: Introduction and Comparison
(with discussion). IMS Lecture Notes-Monograph Series, Ed.
P. Lahiri. Vol. 38, pp. 135-207.(2001)
Berger, J. and Pericchi, L.: Training samples in objective
Bayesian model Selection. Annals of Statistics, 32, 841–869,3
(2004)
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u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Berger, J., Pericchi, L., and Varshavsky, J.: Bayes Factors and
Marginal Distributions in Invariant Situations. Sankya A, 60,
135–321 (1998)
De Santis, F., and Spezzaferri, F.: Alternative Bayes factors
for model selection. Canadian J. Statist., 25, 503–515 (1997)
Dudley, R.M. and Haughton, D.: Information Criteria for
Multiple Data Sets and Restricted Parameters. Statistica
Sinica, 7, 265–284 (1997)
Giron, F.J., Moreno E. and Casella G.:Objective Bayesian
analysis of multiple changepoints models (with discussion). To
appear in Bayesian Statistics 9, Oxford University Press. (2006)
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Huntjens et. al.: Inter-identity amnesia in dissociative identity
disorder: a simulated memory impairment? Psychological
Medicine, 36, 857–863 (2006)
Kato, B.S. and Hoijtink, H.: A Bayesian approach to
inequality constrained linear mixed models: estimation and
model selection. Statistical Modelling, 6, 231-249.(2006)
Klugkist, I., Hoijtink, H.: The Bayes factor for inequality and
about equality constrained models. Computational Statistics
and Data Analysis, 51, 6367-6379. (2007)
Laudy, O., Hoijtink, H.: Bayesian methods for the analysis of
inequality constrained contingency tables. Statistical Methods
in Medical Research, 16, 123-138.(2007)
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses

Outline
Introduction
Coherent Approach
Informative Hypotheses
Illustration: Dissociative Identity Disorder
Conclusions
Acknowledgements
Lempers, F.B.: Posterior Probabilities of Alternative Linear
Models. Rotterdam: University of Rotterdam Press. (1971)
O’Hagan, A.: Fractional Bayes Factors for Model
Comparisons. Journal of the Royal Statistical Society Ser. B,
57, 115–149 (1995)
Pericchi L. R.: Model Selection and Hypothesis Testing based
on Objective Probabilities and Bayes Factors. Handbook of
Statistics, Vol. 25, pp. 115-149. (2005)
Zellner, A., Siow, A.: Posterior odds for selected regression
hypothesis. In: Bernardo J.M. et al (Eds.), Bayesian Statistics,
vol. 1. Valencia University Press, Valencia, pp. 585-603. (1980)
David Torres N´nez Luis Ra´l Pericchi Guerra Guimei Liu ,
u˜ u Objective Bayes Factors for Informed Hypotheses