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8/11/2019 Bayes Theorem for Choquet Capacities
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Carnegie Mellon University
Research Showcase
Department of Statistics Dietrich College of Humanities and Social Sciences
9-1-1990
Bayes' Teorem for Choquet CapacitiesLarry A. WassermanCarnegie Mellon University
Joseph B. KadaneCarnegie Mellon University, [email protected]
Follow this and additional works at: hp://repository.cmu.edu/statistics
Tis Article is brought to you for free and open access by the Dietrich College of Humanities and Social Sciences at Research Showcase. It has been
accepted for inclusion in Department of Statistics by an authorized administrator of Research Showcase. For more information, please contact research-
Recommended CitationWasserman, Larry A. and Kadane, Joseph B., "Bayes' Teorem for Choquet Capacities" (1990).Department of Statistics. Paper 16.hp://repository.cmu.edu/statistics/16
http://repository.cmu.edu/?utm_source=repository.cmu.edu%2Fstatistics%2F16&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://repository.cmu.edu/statistics?utm_source=repository.cmu.edu%2Fstatistics%2F16&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://repository.cmu.edu/hss?utm_source=repository.cmu.edu%2Fstatistics%2F16&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://repository.cmu.edu/statistics?utm_source=repository.cmu.edu%2Fstatistics%2F16&utm_medium=PDF&utm_campaign=PDFCoverPagesmailto:[email protected]:[email protected]://repository.cmu.edu/statistics/16?utm_source=repository.cmu.edu%2Fstatistics%2F16&utm_medium=PDF&utm_campaign=PDFCoverPagesmailto:[email protected]:[email protected]://repository.cmu.edu/statistics/16?utm_source=repository.cmu.edu%2Fstatistics%2F16&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://repository.cmu.edu/statistics?utm_source=repository.cmu.edu%2Fstatistics%2F16&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://repository.cmu.edu/hss?utm_source=repository.cmu.edu%2Fstatistics%2F16&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://repository.cmu.edu/statistics?utm_source=repository.cmu.edu%2Fstatistics%2F16&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://repository.cmu.edu/?utm_source=repository.cmu.edu%2Fstatistics%2F16&utm_medium=PDF&utm_campaign=PDFCoverPages -
8/11/2019 Bayes Theorem for Choquet Capacities
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The
nnals
of Statistics
1990 Vol . 18, No.3
1328-1339
BAYES THEOREM FOR CHOQUET CAPACITIES
By LARRY WASSERMAN AND JOSEPH B.
KADANE
2
Carnegie Mellon University
We give an upper b ou nd for the posterior probability of a measurable
set
when the prior lies in a class
of
probability measures The bound
is a ratio nal fun ctio n
of
two Choquet integrals.
If
g; is weakly compact and
is closed with respect to majorization, then the bound is sharp if and only if
the
upper
prior probability is 2-alternating. The result is u se d to compute
-bounds for sev eral sets of p riors u sed in robust Bayesian inference.
The
result may be regarded as a characterization
of
2-alternating Choquet
capacities.
1.
Introduction. Sets of probability measures arise
naturally
in classical
robustness, Bayesian
robustness and group
decision making. So me sets of
probability measures give rise to
upper
probabilities
that
are 2-alternating
Choquet
capacities. These
upper
probabilities
are
pervasive
in
the
robustness
literature;
see
Huber
1973),
Huber and Strassen
1973),
Buja
1984, 1985,
1986), Bednarski 1982)
and
Rieder 1977), for example.
The purpose of
this
paper
is
to
prove a version
of
Bayes
theorem
for
sets of prior
probabilities
that
have the 2-alternating property and to see h ow these sets of probabilities may
be
exploited in Bayesian robustness.
This result
was p roved
in Wasserman
1 988, 1990) for infinitely
alternating
capacities also
known as
belief func
tions; Shafer, 1976). Since infinitely
alternating
capacities
are
also
2-alternat
ing, the present
result
generalizes
that theorem.
However, the proof in the
infinitely
alternating
case
uses
an argument
that
depends
on
properties
that
infinitely
alternating
capacities possess
that
are not
shared
by 2-alternating
capacities
in
general. Also,
in
this paper,
the
conditions given are both neces
sary and
sufficient. A
proof of
sufficiency
when the parameter
space is finite is
given in Walley 1981).
Section 2
of
this
paper states and
proves the
main
result. In Section 3, we
apply the resul t to derive explicit bounds for the
posterior
probability
of
a
measurable
set
using
various classes
of
priors. Section 4
contains
a discussion.
2. Main resul t. Let 0
be a Polish space, that is, the topology for
0
is
complete, separable
and
metrizable,
and
le t
be the Borel u-algebra for
Let
PI be the set
of
all bounded, nonnegative, real-valued,
measurable
func
tions
defined
on Let { o }
be a
set of
probability
measures on
a
sample space Y
with
u-algebra
N.
Assume that
each
o
has
a d en si ty
{ YI
Received October 1988; revised May 1989.
Research supported by a post-doctoral fellowship from the
Natural
Sciences
and
Engineering
Research Council
of
Canada.
2Research supported by NSF grant DMS-87-05646.
S
1980 subject classifications
Primary
62F15; secondary 62F35.
Key words
and
phrases 2-alternating Choquet capacities, robust Bayesian inference, posterior
bounds.
1328
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8/11/2019 Bayes Theorem for Choquet Capacities
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BAYES THEOREM AND CAPACITIES 1329
with
respect
to
some u-finite dominating
measure
and let
L
=
f YI
be the
likelihood function for
having observed
Y
E
Y.
We
assume
that
LEge
for
each Y
E
Y.
Let 9 be a nonempty
set
of prior probability measures on
and define
the
upper and lower
prior
probability functions by
P
A
=
sup
P A and E
A
= in f P A .
P P
Since and E are r el at ed b y the equation peA = 1 - E AC ,
it
suffices to
concentrate on p. In what follows,
there
is no loss in generality by
assuming
that 9 is convex.
For
each X
E
gc, define
the upper
expectation
of
X by
E X
=
supp
E
g;P X ,
where
P X
=
jX P d .
The
lower expectation
of
X, denoted by
~ X ,
is defined
in an
analogous way. Let g; be
the
class of
posteriors arrived at by applying Bayes theorem to each
prior P
in 9 for
which peL > o. To avoid triviality, we
assume there
is
at
least one
such
P.
Let P
y
be the upper probability generated by g;
The
posterior corresponding
to
P is de note d b y P
y
We say that 9 is 2-alternating if for each
A,
B E ~ ,
peA
u B ~ peA +
PCB -
p A n B .
We say that 9 generates a Choquetcapacity if
p F n ~ p F
for each sequence of closed sets F
n
F. It can be shown that 9 generates a
Choquet capacity if and only if {P; P ~ p} is weakly compact. Here, P ~
means
that peA
~
peA
for every
A
E
~ . See Choquet
1953)
and Huber
and Strassen 1973) for details on capacities. We say
that
9 is closed with
respect to majorization, or is
m-closed, if
P
~
implies that P
E
It is
common practice
in
the literature
to assume
that 9 is m-closed. We will show
Section 3, Ex ample
1
that repla cing 9 with its m-closure has nontrivial
consequences.
For each X E
g
the
upper
Choquet integral of X is defined by
C X
=
[OP X
t
dt,
o
where X
t
{ E X > t}. Similarly, the lower Choquet integral Q X is
defined by
It follows easily that
Q X
~
E X
~ E X ~ C X .
If
9 consists
of
a single
probability measure
P,
then
C X
= Q X = E X = E X = P X ; if this
probability
measure
is our prior probability, then the posterior probability of a
subset
A,
after
observing
y
and
applying Bayes theorem, may be expressed
as
P LA / P L
A
P LAc , where L
A
= L I
A
and I
A
is
the
indicator
function for A. Note that C and Q may b e e xte nde d t o the
set
of all bounded,
mea su ra bl e func tions b y a ddi ng and subtracting a sufficiently large positive
constant.
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1330
L. A WASSERMAN AND J B KADANE
THEOREM Let
g;
a nonempty set
of
prior probabilities on
an d
let
g;
the corresponding class ofposterior probabilities. Then for each A E
when the ratios are well defined. If
g;
generates a Choquet capacity
and
i f
g;
is m-closed then the following three statements are equivalent:
i
P is 2-alternating.
ii The first inequality is
an
equality for each A E
and
each LEge
iii
The second inequality is
an
equality for each A E
an d
each LEge
Before proving
the theorem
we shall
state a few
lemmas The first
lemma is
from
Huber and
Strassen
1973 .
LEMMA 1 If
g;
is m-closed
an d
generates a 2-alternating Choquet capac
ity then for each upper semicontinuous
X
E ge there exists
aPE g;
such that
P X) =
E X)
and
P X
t
=
P X
t
),
for each real number t.
The next
two
lemmas are from Bednarski 1982 .
LEMMA
2.
If
0,
P L
A
PiA = P L
A
P L
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BAYES THEOREM AND CAPACITIES
Now, Py A is
of
the form
1331
1
h P
=
1
g P / f P
where
rep
= P L
A
and g P
= P LAc . h P is maximized where g p jr p
is minimized.
But
which proves
the
first inequality.
The
second inequality follows from
that
fact
that E X ~ C X
and
~ X
~ Q X .
Now suppose
g
is m-closed
and
generates
a
Choquet
capacity. We begin by
showing
that
i) implies ii). Assume
that
P is 2-alternating. We need
to
show
that the first inequality is an equality for each A E
and
for each L E fJ If
E L
A
= 0
this
follows easily. Now suppose
E L
A
>
0
Let b
=
SUPOEACL O .
Define a function a
on
e
by a O =
L O
b on
A
and
a O = -
L O
b on
AC.
For
notational convenience, define
E L
A
B=
E LA ~ L A c .
We will show
that
for ea ch
e
>
0,
there
exists
P
~
P
such
that
P L
A
B e<
0,
there
exists, by
Lemma
4, a nonnegative, bounded,
upper
semicontinuous function h such
that
h
~
a
and E a -
8
~ E h ~
E a .
Set
k
=
E L
A
~ L A c .
Note that
k >
0 Choose 8
t =
P h > t ,
for e ac h
t.
It
is easily verified that
P L
A
~ E L
A
-
8 and P LAc
~ ~ L A c
8. Thus
P L
A
E L
A
8
P
y
A =
P L
A
P LAc E L
A
-
8
E LAc
8
8 8
= B
-
E L
A
~ L A C
=
B
-
k
>
B
-
e.
Since
this
holds for e ac h
e >
0
supP
E gPy A = B so
that
ii) holds.
Further
i) implies iii) because
of Lemma
4.
To prove that ii) implies i), first
assume
that e is finite. Suppose
that
Py A = E LA j E L
A
~ L A c
for every
subset A
and
every
L E fJ
g is
closed,
with
respect
to
the
Euclidean topology, so
there
exists, for e ac h
A
a
PEg
such that Py A = Py A . Therefore, .
E L
A
P L
A
E L
A
~ L A c
P L
A
P LAc ,
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8/11/2019 Bayes Theorem for Choquet Capacities
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1332
L.
A
WASSERMAN AND
J. B
KADANE
from which it follows that
1
This
holds for each
A an d
each
L E [ C. Le t A l C A
2
C C An
be an
increasing sequence
of
subsets.
For
i =
1 2 . . .
n -
1
define
L
i
on to
be
equal to 1 on
Ai an d Ai+l
an d
0 otherwise.
From
1) we deduce
that there
exists
Pi E
9 such that Pi A
i
)
=
P A
i
) an d
Pi A
i
+
l
)
=
P A
i
+
l
)
for i
=
1 2 . . .
n
- 1
Define
A
o
=
0
and
let
gi
=
A i
-
A i - I Let ~
be
th e
field
generated by {gl
,gn}
Define an additive se t function
S
on
~
by
S gl) = P l A
l
),
S g2) =P
2
A
2
) -P l A
l
),
S gn-l) =
P
n
-
l
(
An-I)
- P
n
-
2
( A
n
-
2
),
S gn)
=
Pn-l A
n
) - Pn-l A
n
-
l
)
and S A ~
= 1 -
SCAn)
Then,
S A
i
)
=
P A
i
)
for
i
= 1, 2,
. . .
,
n.
Also,
S gl) ~
o
and S gi)
= Pi A
i
) - Pi-l A
i
-
l
)
=
P A
i
) - Pi-l A
i
-
l
)
=
Pi-l A
i
)
Pi-l A
i
-
l
) ~ 0
so that
S
is a probability
m easure on ~
which maximizes th e
probability
of
the
Ai S
and
since-
S
is dominated by
on ~
S
may be
extended
by
t he Hahn-Ba na ch
theorem
to
2.
By
Lemma
2,
satisfies
th e
2-alternating
condition. Therefore 9 is 2-alternating.
Now we show
that
iii) implies i)
and
we continue
with th e assumption that
is finite. Let th e second inequality be an equality for each A E
~ and
each
L E
[ C. Then
E X ) = C X ) for each bounded function X.
For
suppose E X )
a}. We
claim that
E X
A
)
< C X
A
.
To see this ,
note
that if
a = Q t. en
E X ~
=
E X )
< C X)_= C X
A
. If a > 0 then-py th e
~ i n i t i o n
of C,
C X
A
) = C X ) - aE(AC) >
E X
A
) + Ll where
Ll
= E X ) -
E X
A
)
aE(AC).
Now, for each
P E 9
P X
=
P X
A
)
+
P XAc)
=
P X
A
)
+
aP AC)
~
P X
A
)
+
aE AC),
so
that
E X )
~ E X
A
)
+
aE(AC).
Hence,
Ll
~ 0 which implies that
E X
A
)
b > c > d >nd b > c d. Now let P be a
probability
measure
with values ~ ~ 0, 0 on e and let Q have values
0, o ~ Finally, let 9 be
the
convex hull of P and Q It is easily verified
that
9 is m-closed
and that P
is
not
2-alternating. Let
= {I}.
Then
_ a E L
PiA
=
a
b
and
Q X =
1
-
e P X
e inf X O .
JEE>
Applying
the
theorem, we have
P
A
= 1 - e P i
A
ea/P L
y
l - e +ea jP L
where
a
= sUPo E L O . This result was stated
in
Huber 1973 .
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BAYES THEOREM AND CAPACITIES 1335
EXAMPLE
4. Tot al varia ti on neighbourhoods. As
in
Example 3, fix
and e
and let < be compact. Define by
9= {Q;Ll P,Q)
e},
where Ll P, Q) = SUPAE.98IP A - Q A)I. This is the e t ot al -varia ti on class
generated by
P.
Let
x
V
y
denote the maximum
of x
and
y
and
x
\
y
the
minimum of
x and
y.
Then peA)
= P A)
+ e) \
1 A 0 and
E A)
=
P A) - e) V
A < Again, is m-closed and generates a 2-alternating
Choquet capacity. For every X E f C define to X)
=
sup{t; P X
t
) 1 - e} and
so X)
=
inf{t; P X
t
)
e}. Also, let
Then
C X) =P X) esupX l - e t
o
X)
-J X,O, to X))
and
Q X) =
P X)
+ e
inf X -
eso X)
- J X, so X), sup
X).
Let to = to L
A
) and So = so LAc). Then,
_ P L
A
)
1 -
e)t
o
ea
- J L
A
,
0,
to)
y {
= ) ) ) ,
P L + I - e
t o+ea - J
LA,O,t
o
- e s o - J
LAc,so,b
where a = SUPOEE> LA O and b = SUPOEE> LAc O .
Bednarski 1981) and B uj a 1986) considered an upper probability that
contains e-contamination and
total
variation neighbourhoods as special cases.
It may be shown that this general ized class is an m-closed, 2 alternating
Choquet capacity as well.
EXAMPLE
5
Density bounded classes. Let Land U be two u-finite mea
sures on
such
that
L
U;
that
is,
L A)
U A)
for e ach
A
E
Lavine
1987) considers classes of probability
measures
of the form = {P; L P
U}.
He
calls
this
a
density bounded class.
We shal l
assume
that
L
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8/11/2019 Bayes Theorem for Choquet Capacities
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1336
L. A WASSERMAN AND J. B. KADANE
L X)
-
SO X) d
U
J X,
0,
so X)).
Here, U X)
= jX O)U dO) and L X) =
jX O)L dO). Therefore,
- U L
A
) tod
L
-
J L
A
,
0,
to)
P
y
A
=
U LA)
to d
L
- J LA
0,
to)
L LAc)
-
So d
u
J
L c 0,
so)
where
to
to L
A
) and So
so LAc).
EXAMPLE 6 Probabilities with
partition
restr ictions. Moreno
and
Cano
1988 consider
the
following
set of
priors.
Let P
be a fixed probability
measure on yg
e
a fixed
number in
[0,1]
and H
=
{hI , h
k
}
a measurable
partition
of
8
Assume
that
8 is compact. Let P = PI
,Pk)
be such
that
each
Pi
nd
LPi
=
1
Then
define
9= {Q;Q
= 1 -
e)P
e8,
8
E
J} .
Here
is
the set
of all probability measures that satisfy 8 h
i
) = Pi i =
1 2
. . .
k. The interpretation
is
the
same as
the
e-contaminated class
of
Example 3, except that we are
putting
constraints on the possible contamina
tions by forcing
them
to have fixed probabilities on some part it ion. This is a
simple, reasonable way to restrict the size
of the
possible contaminating
distributions.
This class contains two
important
special cases. By
taking
H =
8
we get
back
the
general e-contaminated model
of
Example 3 By
taking
e
=
1 we
get
the set of
all probability measures with fixed probabilities on a par ti tion. This
latter class has been studied by Berliner
and
Goel 1986 .
Define
A
=
{i; hi c
A}
and J
A
=
{i; hi
n A 0}
Then
f A = 1 - e)P A) e L Pi
iE I
A
and
P A) = 1 -
e)P A) e L Pi.
iE J
A
It
may
be shown
that
P
is
n alternating
for every
n and
hence is 2-alternat
ing.
Further 9
generates an m-closed, Choquet capacity. It follows
that
Q X)
=
1 -
e)P X)
eL KiPi
and
where Ki = n
OEht
X O) and
Xi
= sUPOEh
t
X O).
Therefore,
- 1 -
e)P L
A
) eLPiLA.
PiA
=
1 -
e)P L)
eLPi[
L
Ai
i
EXAMPLE 7
Symmetric, unimodal contaminations. Sivaganesan
and
Berger
1989 consider
the
following modification
of the
e-contaminated class for 8
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BAYES THEOREM AND CAPACITIES
equal to the real line. For fixed
P
and
set
g;
= {Q = 1 - )
P
S , S
E
J }
1337
where
J
is
the
se t of all probability measures
that
possess densities
that
are
unimodal
and
symmetric about a fixed point Jo.
Without
loss
of
generality, we
take
0
0
= 0. Now, g; generates a 2-alternating upper probability if and only if
J generates
a 2-alternating upper probability. Hence, we confine attention to
the
class J . As Sivaganesan
and
Berger (1989) remark
when
computing
extrema, one need only consider the smaller class consisting of
uniform
probability measures
that
are centered at the origin. A
point
mass at the origin
must
be
included
as
well. Let
S be the
upper probability generated by
J .
We
claim
that
S
is
not
2-alternating
and
that
J
is
not
m-closed.
To see that S is not 2-alternating,
set
A = [- 3, - 1] U [1,2]
and
B
=
[-2 - 1] U [1,3].
Then
i =
S A
U
B) >
S A) S B) -
S A n B)
= ~
~ - To see that J is
not
m-closed, let
R
= ~ o
~ u
where is a point
mass at
nd
u is the uniform [0 1] distribution. Now, if
A, R A) 1 =
S A). Otherwise, R A) = R A
n
[0,1]) = u A
n
[0 1] /2 =
u A n
[0, 1]
u A) S A), where u is uniform [
1
1]. Thus R S but R . J so J is
not m-closed.
As a result, the Choquet integral fails miserably
as
an approximation to
For
example,
let
X
n
be defined by
0
X
n
= n
n
i f n ~ O ~ n
if 0 > n
i fO < n .
Then E X
n
= E X
n
= 0, but it is easily shown that C X
n
= n/2 Hence,
limn
-+oo(C(X
n
) - E X
n
=
00 .
This is an
extreme
example
that
shows how
important the 2-alternating and m-closure conditions are. It would
be
interest
ing to investigate how much of the difference between E and
C
is accounted
for
by
the failure
of
the
2-alternating
condition
and
how much is because
of
the
lack
of
m-closure.
4. Discussion. The main result of
this
paper helps to unify
the
process of
finding bounds for posterior probabilities in robust Bayesian inference. Also,
this
theorem may
be
thought of as
a characterization
of 2-alternating
Choquet
capacities.
Other authors
have given different characterizations.
Huber and
Strassen (1973) showed that the 2-alt
/
ernating
structure is necessary
and
sufficient for generalizing the Neyman-Pearson lemma to
sets
of probabilities.
Buja (1984, 1985) generalized
this
result from the case of two hypotheses
to
a
finite
number of
hypotheses. Similarly, Bednarski (1982) showed that this
condition is necessary
and
sufficient for
the
construction
of
least in(ormative
binary experiments in
the
sense of Blackwell (1951), Le
Cam
(1964, 1969,
1972)
and
Torgersen (1970). These
authors
were concerned
with
uncertainty
with respect to sampling distributions, as
represented
by sets
of
probability
measures dominated
by 2-alternating
Choquet
capacities. We have focused on
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1338
L.
A
WASSERMAN AND J. B KADANE
uncertainty in
the
prior distribution. Other properties of 2 alternating capaci
ties may be found in Walley 1981).
Acknowledgments.
We
are grateful to the
referees for
their
insightful
comments and suggestions.
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