Bayes Theorem for Choquet Capacities

8/11/2019 Bayes Theorem for Choquet Capacities

1/13

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]

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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
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2/13

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


3/13

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.


4/13

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


5/13

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 ,


6/13

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 .


9/13

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


10/13

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


11/13


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


12/13

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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DEPARTMENT OF STATISTICS

232

BAKER

HALL

CARNEGIE MELLON UNIVERSITY

PITTSBURGH PENNSYLVANIA

15213-3890

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Bayes Theorem for Choquet Capacities

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