SET-BASED

BAYESIANISM

Henry

E.

Kyburg

Jr.

Michael

Pittarelli

SET-BASED

BAYESIANISM

ABSTRACT

Problems

for

strict

and

convex

Bayesianism

are

discussed

A

set-based

Bayesianism

generalizing

convex

Bayesianism

and

intervalism

is

proposed

This

approach

abandons

not

only

the

strict

Bayesian

requirement

of

a

unique

real-valued

probability

function

in

any

decision-making

context

but

also

the

requirement

of

convexity

for

a

set-based

representation

of

uncertainty

Levi

's

E-admissibility

decision

criterion

is

retained

and

is

shown

to

be

applicable

in

the

non-convex

case

Introduction

The

reigning

philosophy

of

uncertainty

representation

is

strict

Bayesianism

One

of

its

central

principles

is

that

an

agent

must

adopt

a

single

real-valued

probability

function

over

the

events

recognized

as

relevant

to

a

given

problem

Prescriptions

for

defining

such

a

function

for

a

given

agent

in

a

given

situation

range

from

the

extreme

personalism

of

deFinetti

-LRB-

-RRB-

and

Savage

-LRB-

-RRB-

to

the

objective

Bayesianism

of

Jeffreys

-LRB-

-RRB-

and

Jaynes

-LRB-

-RRB-

The

latter

in

the

form

of

Jaynes

maximum

entropy

principle

-LRB-

-RRB-

is

in

favor

among

researchers

in

artificial

intelligence

-LRB-

Cheeseman

-RRB-

The

appeal

to

computer

scientists

of

maximum

entropy

methods

is

partly

explained

by

the

relative

ease

with

which

they

permit

the

computation

of

unambiguous

solutions

to

inference

and

decision

problems

faced

by

artificial

and

interactive

systems

Criticism

of

the

restriction

of

an

agent

to

a

single

probability

function

dates

back

at

least

to

Keynes

-LRB-

-RRB-

More

recently

Levi

-LRB-

-RRB-

and

Kyburg

-LRB-

-RRB-

have

developed

alternatives

to

strict

Bayesianism

that

have

been

dubbed

convex

Bayesianism

and

intervalism

respectively

Levi

's

convex

Bayesianism

allows

an

agent

a

set

of

probability

functions

but

the

set

must

be

convex

He

has

developed

a

decision

method

that

reduces

to

strict

Bayesian

maximization

of

expected

utility

relative

to

a

single

probability

function

when

the

set

of

functions

has

only

one

member

Kyburg

's

approach

retains

a

single

probability

function

but

allows

the

function

to

be

interval-valued

the

probability

of

an

event

is

not

a

single

number

but

an

interval

of

numbers

Mathematically

the

proposals

are

similar

Seidenfeld

-LRB-

-RRB-

has

argued

that

for

purposes

of

decision-making

the

interval-valued

function

should

be

replaced

with

the

-LRB-

convex

-RRB-

set

of

real-valued

functions

bounded

by

the

intervals

Levi

's

methods

may

then

be

applied

to

this

set

The

leading

alternative

to

strict

Bayesianism

among

practitioners

is

Dempster-Shafer

Theory

in

which

non-additive

belief

functions

replace

classical

probability

functions

-LRB-

Shafer

-RRB-

Kyburg

-LRB-

-RRB-

has

shown

that

although

this

theory

is

a

generalization

of

strict

Bayesianism

it

is

a

special

case

of

convex

Bayesianism

For

any

belief

function

there

exists

a

convex

set

of

classical

probability

functions

defined

on

its

frame

of

discernment

from

which

it

can

be

recovered

However

there

exist

-LRB-

perfectly

meaningful

and

reasonable

-RRB-

convex

sets

of

classical

probability

functions

to

which

there

do

not

correspond

belief

functions

We

propose

to

extend

convex

Bayesianism

and

intervalism

to

a

more

general

set-based

Bayesianism

Set-based

Bayesianism

would

drop

not

only

the

strict

Bayesian

requirement

of

a

unique

real-valued

probability

function

but

also

the

requirement

of

convexity

for

a

set

of

functions

The

approach

would

be

Bayesian

by

-LRB-

among

other

things

-RRB-

retaining

the

requirement

that

a

decision

method

for

sets

of

probability

functions

reduce

for

unit

sets

to

maximization

of

expected

utility

Problems

for

Strict

Bayesianism

We

use

the

term

strict

Bayesianism

following

Levi

-LRB-

-RRB-

According

to

the

strict

Bayesian

view

a

rational

agent

is

committed

to

recognizing

a

single

probability

function

for

use

in

computing

expected

utilities

in

any

given

context

of

deliberation

and

is

compelled

to

select

a

course

of

action

that

has

maximum

expected

utility

relative

to

that

probability

function

among

the

alternatives

under

consideration

There

are

many

types

of

information

bearing

on

an

algebra

of

events

relative

to

which

a

strict

Bayesian

would

be

hard-pressed

to

come

up

with

a

classical

numerically

determinate

probability

function

A

number

of

these

situations

are

discussed

in

the

following

sections

In

some

of

the

examples

further

reflection

-LRB-

or

badgering

-RRB-

to

reduce

the

indeterminacy

seems

to

be

an

implausible

solution

In

others

there

is

no

further

information

that

could

be

-LRB-

or

could

have

been

-RRB-

obtained

that

would

help

the

agent

arrive

at

a

single

probability

function

The

method

of

maximum

entropy

by

which

the

indeterminacy

is

resolved

algorithmically

-LRB-

and

hence

objectively

-RRB-

is

also

discussed

and

criticized

Frequencies

Many

of

the

probabilities

that

we

set

the

greatest

store

by

and

that

we

regard

as

the

most

reliable

bases

for

decision

in

the

face

of

uncertainty

are

probabilities

that

are

based

on

statistical

knowledge

Thus

we

think

we

know

that

ordinary

coins

land

heads

nearly

half

the

time

that

the

dice

in

well

regulated

casinos

land

with

double-one

about

of

the

time

that

the

frequency

of

males

among

newborn

humans

is

very

close

to

and

so

on

and

when

possible

we

would

like

to

base

the

probabilities

that

we

use

for

making

decisions

on

such

statistical

knowledge

Despite

the

worry

that

we

often

have

no

statistics

relevant

statistical

data

can

often

be

found

The

probability

of

heads

on

the

one

and

only

toss

of

a

newly

minted

coin

that

is

then

to

be

destroyed

can

perfectly

well

be

based

on

the

masses

of

statistics

we

are

assumed

to

have

concerning

coin

tosses

in

general

Standard

statistical

methods

yield

confidence

intervals

or

lead

to

the

rejection

of

all

but

a

set

of

hypotheses

as

the

examples

cited

indicate

There

are

two

difficulties

for

the

strict

Bayesian

First

the

strict

Bayesian

can

not

tolerate

an

interval

he

needs

a

real

number

in

order

to

be

strict

Second

classical

statistical

methods

are

themselves

not

acceptable

for

the

strict

Bayesian

The

strict

Bayesian

can

not

identify

a

probability

with

an

interval

he

is

committed

to

choosing

some

point

presumably

a

point

in

that

interval

to

represent

his

degree

of

belief

But

there

seems

no

non-arbitrary

way

to

choose

a

point

The

midpoint

makes

sense

only

if

the

distribution

of

belief

in

the

interval

is

symmetric

Furthermore

from

a

Bayesian

perspective

even

if

the

midpoint

were

the

maximum

likelihood

estimate

of

the

parameter

in

question

the

probability

that

the

parameter

had

that

value

would

be

zero

The

Bayesian

statistician

does

not

have

the

problem

of

choosing

a

point

in

that

interval

because

he

rejects

classical

methods

He

insists

that

all

inference

be

done

through

Bayes

theorem

and

that

one

must

always

have

a

prior

distribution

implicitly

if

not

explicitly

in

order

to

perform

inference

This

means

that

the

correct

characterization

of

my

belief

about

ordinary

coins

is

that

the

proportion

of

heads

that

they

yield

can

be

anywhere

from

to

with

a

sharp

high

peak

of

belief

in

the

neighborhood

of

a

half

In

order

to

make

a

decision

based

on

my

beliefs

about

coins

and

their

flips

I

must

integrate

to

find

the

probability

of

heads

pF

-LRB-

p

-RRB-

dp

Prob

-LRB-

Heads

-RRB-

It

is

true

that

the

value

of

this

integral

is

a

half

or

close

to

it

but

it

is

not

clear

that

we

must

start

with

a

prior

distribution

of

heads

on

tosses

of

coins

in

advance

of

any

evidence

It

is

at

any

rate

not

clear

that

classical

statistical

methods

should

be

rejected

out

of

hand

together

with

the

confidence

intervals

and

families

of

unrejected

hypotheses

that

accompany

them

Qualitative

comparative

probability

A

statement

of

qualitative

comparative

probability

for

example

A

is

more

probable

than

B

may

be

the

strongest

statement

that

a

person

is

willing

to

make

One

might

press

such

a

person

for

a

single

number

for

the

probability

of

A

by

iteratively

quizzing

him

on

preference/indifference

between

a

lottery

with

prize

X

and

probability

p

-LRB-

A

-RRB-

of

winning

and

a

lottery

with

prize

X

and

some

stated

probability

of

winning

-LRB-

initially

say

if

B

is

not

A

corresponding

to

the

statement

A

is

more

likely

than

not

-RRB-

until

convergence

to

p

-LRB-

A

-RRB-

is

achieved

with

sufficient

precision

Of

course

an

individual

may

as

part

of

the

introspective

process

of

assessing

his

subjective

probabilities

apply

the

procedure

to

himself

Such

a

procedure

whether

it

is

applied

introspectively

or

by

a

bullying

psychologist

may

not

be

successful

The

person

may

express

indifference

between

a

lottery

with

prize

X

and

probability

p

-LRB-

A

-RRB-

of

winning

and

any

number

of

lotteries

with

prize

X

and

stated

probabilities

of

winning

between

and

When

it

comes

to

buying

and

selling

lotteries

of

course

he

will

be

consistent

But

we

have

little

reason

to

take

very

seriously

the

real

number

he

happens

to

settle

on

in

one

instance

of

the

measurement

procedure

Since

probabilities

are

assessed

not

for

their

own

sake

but

to

guide

action

there

may

be

little

point

in

conducting

many

cycles

of

inquiry

or

introspection

There

may

only

be

limited

evidence

yielding

broad

bounds

on

the

probabilities

and

yet

this

may

suffice

perfectly

well

for

decision

Example

A

woman

has

taken

her

ill

toddler

to

a

country

doctor

and

reported

that

the

child

has

swallowed

either

substance

V

or

substance

R

but

that

it

is

more

likely

that

R

was

swallowed

-LRB-

The

child

was

left

unattended

near

open

bottles

of

both

after

drinking

any

of

one

the

child

would

not

drink

any

of

the

other

both

bottles

are

on

their

sides

both

liquids

are

odorless

but

R

is

red

and

V

is

colorless

the

mother

is

aware

of

studies

indicating

that

children

prefer

colorful

things

and

has

no

reason

to

believe

that

her

child

is

an

exception

-RRB-

There

are

no

signs

-LRB-

e.g.

dilated

pupils

-RRB-

that

would

distinguish

between

the

two

substances

that

can

be

detected

without

equipment

unavailable

to

the

doctor

Ipecac

is

the

suggested

treatment

for

ingestion

of

V

rest

is

recommended

for

R-poisoning

The

doctor

has

two

alternatives

First

he

may

attempt

to

determine

the

exact

probability

that

the

child

has

swallowed

R

by

relentlessly

quizzing

the

mother

Would

you

he

might

demand

prefer

a

lottery

with

a

prize

X

and

probability

p

-LRB-

R

-RRB-

where

p

-LRB-

R

-RRB-

is

the

probability

that

the

child

swallowed

R

to

a

lottery

with

prize

X

and

chance

of

winning

No

Well

what

about

a

lottery

The

other

alternative

is

to

ignore

the

exact

magnitude

of

the

mother

's

belief

-LRB-

Why

should

this

be

relevant

anyway

-RRB-

Without

further

delay

the

doctor

is

able

to

infer

bounds

on

p

-LRB-

V

-RRB-

and

p

-LRB-

R

-RRB-

of

p

-LRB-

R

-RRB-

-LSB-

-RSB-

and

p

-LRB-

V

-RRB-

-LRB-

R

-RRB-

-LSB-

-RSB-

Treatment

with

ipecac

is

strongly

contraindicated

in

cases

of

R-poisoning

It

is

not

necessary

to

give

ipecac

in

cases

of

V-poisoning

although

it

will

make

the

child

feel

better

much

sooner

On

this

basis

the

doctor

constructs

the

utility

matrix

R

V

ipecac

rest

The

set

of

classical

probability

functions

compatible

with

the

bounds

may

be

represented

as

a

line

segment

in

the

space

-LSB-

-RSB-

with

endpoints

-LRB-

-RRB-

and

-LRB-

-RRB-

where

the

first

of

the

two

dimensions

represents

p

-LRB-

R

-RRB-

The

doctor

is

aware

that

expected

utility

is

a

linear

function

of

the

probabilities

e.g.

eu

-LRB-

rest

p

-RRB-

u

-LRB-

rest

R

-RRB-

p

-LRB-

R

-RRB-

u

-LRB-

rest

V

-RRB-

p

-LRB-

V

-RRB-

p

-LRB-

R

-RRB-

p

-LRB-

V

-RRB-

and

thus

that

the

maximum

and

minimum

values

for

the

expected

utility

of

either

action

as

p

ranges

over

the

line

segment

of

possible

values

compatible

with

the

qualitative

information

given

by

the

mother

are

achieved

for

one

of

the

endpoints

eu

-LRB-

ipecac

-LRB-

-RRB-

-RRB-

eu

-LRB-

ipecac

-LRB-

-RRB-

-RRB-

eu

-LRB-

rest

-LRB-

-RRB-

-RRB-

eu

-LRB-

rest

-LRB-

-RRB-

-RRB-

Thus

for

any

compatible

classical

probability

function

rest

will

have

the

higher

expected

utility

and

is

prescribed

by

the

doctor

Representing

the

probabilities

as

a

real-valued

function

e.g.

p

-LRB-

R

-RRB-

p

-LRB-

V

-RRB-

will

give

the

same

result

an

advantage

of

the

analysis

using

bounds

is

that

the

doctor

can

be

sure

-LRB-

for

his

utility

matrix

-RRB-

that

prescribing

rest

is

the

better

action

for

any

p

for

which

p

-LRB-

R

-RRB-

More

generally

a

numerical

but

not

necessarily

determinate

representation

can

be

found

for

an

arbitrary

consistent

system

of

qualitative

probability

judgements

Such

a

system

is

consistent

if

a

real

number

can

be

associated

with

each

proposition

so

that

the

classical

probability

axioms

are

satisfied

and

so

that

if

x

is

associated

with

X

and

y

is

associated

with

Y

and

the

statement

X

is

at

least

as

probable

as

Y

is

in

the

system

or

is

implied

by

it

then

x?y

Suppose

that

the

judgments

A

is

at

least

as

probable

as

B

and

B

or

C

is

at

least

as

probable

as

A

are

made

for

mutually

exclusive

and

exhaustive

events

A

B

and

C.

Any

of

the

infinitely

many

solutions

to

the

system

of

linear

inequalities

p

-LRB-

A

-RRB-

p

-LRB-

B

-RRB-

p

-LRB-

C

-RRB-

p

-LRB-

A

-RRB-

p

-LRB-

B

-RRB-

p

-LRB-

B

-RRB-

p

-LRB-

C

-RRB-

p

-LRB-

A

-RRB-

for

example

p

-LRB-

A

-RRB-

p

-LRB-

B

-RRB-

p

-LRB-

C

-RRB-

is

compatible

with

these

judgments

If

nothing

stronger

than

these

comparisons

is

forthcoming

then

there

is

no

basis

for

choosing

a

single

one

of

these

functions

as

representative

of

the

probability

information

On

the

other

hand

statements

of

noncomparative

qualitative

probability

e.g.

A

is

very

probable

are

notoriously

vague

-LRB-

The

decision

of

the

U.S.

to

invade

Cuba

in

has

been

blamed

in

part

on

the

misunderstanding

of

such

a

probability

assessment

-LRB-

Zwick

et

al.

-RRB-

-RRB-

If

a

decision

of

any

consequence

must

be

made

contingent

on

the

occurrence

or

nonoccurrence

of

A

then

an

attempt

should

be

made

to

narrow

the

range

of

possible

numbers

for

p

-LRB-

A

-RRB-

But

it

is

not

necessary

to

determine

a

single

number

as

the

example

above

illustrates

Conditional

probability

assessments

One

may

be

able

to

come

up

with

real

number

values

only

for

certain

conditional

probabilities

Each

assessment

p

-LRB-

E

E

-RRB-

x

is

a

constraint

on

a

classical

probability

function

over

i

j

events

E

in

the

form

of

a

linear

equation

k

p

-LRB-

w

-RRB-

x

p

-LRB-

w

-RRB-

S

S

w?E

E

w?E

i

j

j

on

quantities

p

-LRB-

w

-RRB-

where

w

is

an

elementary

event

In

general

there

will

be

infinitely

many

probability

functions

consistent

with

a

set

of

conditional

probability

assessments

Example

For

a

sample

space

-LCB-

A

B

C

D

-RCB-

suppose

that

p

-LRB-

-LCB-

A

B

-RCB-

-LCB-

B

C

D

-RCB-

-RRB-

and

p

-LRB-

D

-LCB-

A

C

D

-RCB-

-RRB-

are

given

The

former

translates

to

the

constraint

p

-LRB-

B

-RRB-

p

-LRB-

-LCB-

B

C

D

-RCB-

-RRB-

which

implies

p

-LRB-

B

-RRB-

-LRB-

-LCB-

B

C

D

-RCB-

-RRB-

-LRB-

p

-LRB-

B

-RRB-

p

-LRB-

C

-RRB-

p

-LRB-

D

-RRB-

-RRB-

since

B

C

and

D

are

elementary

events

The

pair

of

assessments

thus

translates

into

the

system

of

simultaneous

linear

equations

p

-LRB-

A

-RRB-

p

-LRB-

D

-RRB-

p

-LRB-

B

-RRB-

p

-LRB-

C

-RRB-

p

-LRB-

D

-RRB-

p

-LRB-

A

-RRB-

p

-LRB-

C

-RRB-

p

-LRB-

D

-RRB-

This

system

of

equations

has

infinitely

many

solutions

However

by

linear

programming

it

may

be

determined

e.g.

that

p

-LRB-

D

-RRB-

-LSB-

-RSB-

Marginal

probabilities

Probabilities

on

a

set

of

events

of

interest

must

sometimes

be

inferred

from

other

probabilities

In

certain

cases

the

axioms

of

probability

allow

unambiguous

determination

of

the

values

in

question

while

in

others

they

do

not

For

example

from

p

-LRB-

A

and

B

-RRB-

x

and

p

-LRB-

A

and

B

-RRB-

y

it

may

be

deduced

that

p

-LRB-

A

-RRB-

x

y

but

the

inference

in

the

other

direction

is

not

deductive

Unless

p

-LRB-

A

-RRB-

there

are

infinitely

many

pairs

of

values

for

the

more

elementary

events

compatible

with

its

value

These

pairs

form

a

line

segment

in

the

space

-LSB-

p

-LRB-

A

-RRB-

-RSB-

with

endpoints

-LRB-

p

-LRB-

A

-RRB-

-RRB-

and

-LRB-

p

-LRB-

A

-RRB-

-RRB-

Example

The

movements

of

a

flatfish

in

a

tank

are

being

monitored

by

two

children

enrolled

in

a

summer

program

for

gifted

elementary

school

students

Each

of

the

three

dimensions

of

the

tank

is

bisected

to

produce

eight

regions

The

tank

is

placed

on

a

table

and

the

students

are

seated

so

that

they

face

different

planes

Working

together

for

two

five-day

weeks

in

two

four-hour

shifts

per

day

each

student

records

the

location

of

the

fish

within

the

plane

-LRB-

s

-RRB-

he

is

monitoring

every

twenty

seconds

at

the

sound

of

a

buzzer

Probabilities

-LRB-

based

on

joint

observations

-RRB-

are

estimated

as

X

Y

r

X

Z

r

XY

XZ

L

L

L

L

L

U

L

U

U

L

U

L

U

U

U

U

Each

of

these

numbers

is

a

constraint

on

a

possible

function

p

For

example

r

-LRB-

LL

-RRB-

XYZ

XY

implies

that

p

-LRB-

LLL

-RRB-

p

-LRB-

LLU

-RRB-

XYZ

XYZ

There

are

infinitely

many

classical

probability

functions

p

satisfying

all

eight

of

the

XYZ

-LRB-

linear

-RRB-

equations

p

-LRB-

LLL

-RRB-

p

-LRB-

LLU

-RRB-

XYZ

XYZ

p

-LRB-

LUL

-RRB-

p

-LRB-

LUU

-RRB-

XYZ

XYZ

p

-LRB-

ULL

-RRB-

p

-LRB-

ULU

-RRB-

XYZ

XYZ

p

-LRB-

UUL

-RRB-

p

-LRB-

UUU

-RRB-

XYZ

XYZ

p

-LRB-

LLL

-RRB-

p

-LRB-

LUL

-RRB-

XYZ

XYZ

p

-LRB-

LLU

-RRB-

p

-LRB-

LUU

-RRB-

XYZ

XYZ

p

-LRB-

ULL

-RRB-

p

-LRB-

UUL

-RRB-

XYZ

XYZ

p

-LRB-

ULU

-RRB-

p

-LRB-

UUU

-RRB-

XYZ

XYZ

simultaneously

The

first

four

equations

are

associated

with

the

function

r

-LRB-

and

jointly

XY

imply

p

-LRB-

x

-RRB-

-RRB-

The

set

of

solutions

to

these

four

equations

is

the

extension

-LRB-

Pittarelli

XYZ

Sx

-RRB-

of

r

to

the

space

XYZ

each

member

of

the

extension

has

r

as

its

marginal

distribuXY

XY

tion

over

XY

Similarly

for

the

second

four

equations

and

r

The

extension

of

the

pair

of

XZ

functions

-LRB-

i.e.

the

set

of

solutions

to

the

full

system

of

equations

-RRB-

is

the

intersection

of

their

individual

extensions

Had

a

third

student

at

the

risk

of

being

detected

by

the

fish

and

disrupting

its

natural

pattern

of

movement

been

suspended

above

the

tank

to

record

location

in

the

YZ

plane

-LRB-

parallel

to

the

floor

and

ceiling

-RRB-

the

resulting

set

of

functions

would

have

been

a

subset

but

not

necessarily

a

unit

set

the

function

p

might

remain

indeterminate

Suppose

that

the

obserXYZ

vations

of

the

third

student

resulted

in

the

probability

function

Y

Z

rYZ

L

L

L

U

U

L

U

U

is

compatible

with

r

and

r

in

the

sense

that

the

intersection

of

its

extension

to

XYZ

r

XY

XZ

YZ

with

the

extensions

of

r

and

r

is

nonempty

i.e.

the

system

of

equations

resulting

from

XY

XZ

the

addition

of

the

equations

p

-LRB-

LLL

-RRB-

p

-LRB-

ULL

-RRB-

XYZ

XYZ

p

-LRB-

LLU

-RRB-

p

-LRB-

ULU

-RRB-

XYZ

XYZ

p

-LRB-

LUL

-RRB-

p

-LRB-

UUL

-RRB-

XYZ

XYZ

p

-LRB-

LUU

-RRB-

p

-LRB-

UUU

-RRB-

XYZ

XYZ

to

the

system

of

eight

linear

equations

above

has

a

solution

In

fact

it

has

a

unique

solution

X

Y

Z

pXYZ

L

L

L

L

L

U

L

U

L

L

U

U

U

L

L

U

L

U

U

U

L

U

U

U

However

another

compatible

probability

function

over

YZ

is

Y

Z

rYZ

L

L

L

U

U

L

U

U

There

are

infinitely

many

solutions

to

the

system

of

equations

that

it

determines

Bounds

on

values

of

p

may

be

deduced

from

this

system

as

XYZ

X

Y

Z

pXYZ

L

L

L

-LSB-

-RSB-

L

L

U

-LSB-

-RSB-

L

U

L

-LSB-

-RSB-

L

U

U

-LSB-

-RSB-

U

L

L

-LSB-

-RSB-

U

L

U

-LSB-

-RSB-

U

U

L

-LSB-

-RSB-

U

U

U

-LSB-

-RSB-

Additional

constraints

on

the

events

in

the

space

XYZ

could

be

imposed

together

with

which

the

probability

functions

r

and

r

determine

a

single

numerically

determinate

proXY

XZ

bability

function

p

For

example

under

the

assumption

that

location

along

the

Z

dimenXYZ

sion

is

conditionally

independent

of

location

along

the

Y

dimension

given

the

location

of

the

fish

along

the

X

dimension

in

which

case

p

-LRB-

XYZ

-RRB-

p

-LRB-

XY

-RRB-

p

-LRB-

Z

XY

-RRB-

p

-LRB-

XY

-RRB-

p

-LRB-

Z

X

-RRB-

p

-LRB-

XY

-RRB-

p

-LRB-

XZ

-RRB-

p

-LRB-

X

-RRB-

A

determinate

p

is

calculated

as

XYZ

p

-LRB-

xyz

-RRB-

r

-LRB-

xy

-RRB-

r

-LRB-

xz

-RRB-

-LRB-

r

-LRB-

xz

-RRB-

r

-LRB-

xz

-RRB-

-RRB-

XYZ

XY

XZ

XZ

XZ

where

if

z

L

then

z

U

and

if

z

U

then

z

L

X

Y

Z

pXYZ

L

L

L

L

L

U

L

U

L

L

U

U

U

L

L

U

L

U

U

U

L

U

U

U

But

what

would

be

the

basis

for

such

an

assumption

Of

the

two

independence

relations

that

can

be

tested

against

the

data

one

-LRB-

X

independent

of

Y

-RRB-

holds

exactly

but

the

other

-LRB-

X

independent

of

Z

-RRB-

does

not

even

approximately

r

-LRB-

LU

-RRB-

-LRB-

r

-LRB-

LL

-RRB-

r

-LRB-

LU

-RRB-

-RRB-

-LRB-

r

-LRB-

LU

-RRB-

r

-LRB-

UU

-RRB-

-RRB-

XZ

XZ

XZ

XZ

XZ

Maximum

entropy

methods

Resolution

of

indeterminacy

by

selection

of

the

maximum

entropy

element

of

the

set

of

classical

probability

functions

consistent

with

available

information

is

widely

practiced

Justification

given

for

this

method

ranges

from

the

ease

with

which

the

maximum

entropy

element

is

computed

in

most

cases

to

the

arguments

of

E.T.

Jaynes

-LRB-

-RRB-

that

this

element

is

the

most

likely

to

arise

and

that

it

is

the

least

biased

-LRB-

in

a

non-statistical

sense

-RRB-

of

these

functions

The

last

of

these

the

argument

that

the

maximum

entropy

element

of

such

a

set

embodies

no

more

information

than

is

minimally

implied

by

the

given

constraints

is

easily

dispensed

with

If

information

is

quantified

as

negative

entropy

then

it

is

tautologous

that

adoption

of

the

maximum

entropy

distribution

is

adoption

of

the

distribution

that

is

minimally

informative

among

all

those

compatible

with

the

given

data

But

the

axioms

leading

to

the

form

p

-LRB-

w

-RRB-

log

p

-LRB-

w

-RRB-

k

Sw

-LRB-

i.e.

to

Shannon

's

entropy

-RRB-

for

a

measure

of

uncertainty

defined

for

classical

probability

functions

are

controversial

-LRB-

Macqueen

et

al.

-RRB-

And

intuitively

ruling

out

any

member

of

a

set

of

probability

functions

compatible

with

given

constraints

seems

to

be

going

beyond

the

information

given

Information

measures

for

sets

of

probability

functions

have

been

proposed

-LRB-

Dalkey

Jeffrey

-RRB-

Any

such

measure

will

have

the

property

that

the

information

content

of

a

set

exceeds

that

of

any

of

its

supersets

Thus

a

set

of

distributions

will

have

lower

information

content

than

the

singleton

subset

consisting

of

its

maximum

entropy

element

The

impression

that

many

users

of

maximum

entropy

methods

have

that

the

maximum

entropy

distribution

is

the

most

likely

to

arise

is

due

to

Jaynes

concentration

theorem

-LRB-

-RRB-

Suppose

that

an

experiment

with

a

finite

number

n

of

possible

outcomes

-LRB-

e.g.

die

toss

-RRB-

the

relative

frequencies

of

which

are

subject

to

given

linear

constraints

e.g.

as

implied

by

given

expected

values

-LRB-

E

-LRB-

X

-RRB-

-RRB-

or

marginal

probabilities

-LRB-

p

-LRB-

X

is

odd

-RRB-

-RRB-

is

to

be

repeated

N

times

Suppose

that

experimental

outcome

i

is

observed

f

times

Then

the

number

i

of

possible

sequences

of

outcomes

giving

a

frequency

distribution

f

f

is

given

by

the

n

multinomial

coefficient

N

M

f

f

f

n

Let

p

f

N

Then

i

i

N

M

Np

Np

Np

n

The

relative

frequency

distribution

-LRB-

p

p

-RRB-

with

the

greatest

number

of

possible

n

sequences

of

outcomes

giving

rise

to

it

is

the

distribution

maximizing

M

-LRB-

subject

to

the

constraints

-RRB-

But

maximizing

M

is

equivalent

to

maximizing

any

monotonic

increasing

function

of

M

e.g.

the

function

N

log

M.

But

from

the

Stirling

approximation

-LRB-

log

x

fixlog

x

as

k

k

k

xfi

-RRB-

it

follows

that

as

Nfi

n

N

log

M

fi

p

log

p

k

i

k

i

Si

the

entropy

of

p.

So

the

relative

frequency

distribution

maximizing

entropy

is

the

distribution

with

the

greatest

number

of

sequences

of

experimental

outcomes

corresponding

to

it

Further

-LRB-

and

the

proof

of

this

-LRB-

Jaynes

-RRB-

follows

Pearson

's

derivation

of

the

chi-squared

distribution

-RRB-

as

Nfi

the

closer

-LRB-

as

measured

by

difference

in

entropy

-RRB-

a

relative

frequency

distribution

compatible

with

the

constraints

is

to

the

-LRB-

unique

-RRB-

maximum

entropy

distribution

in

the

set

of

compatible

distributions

the

more

sequences

of

experimental

outcomes

there

are

that

could

generate

it

One

may

object

that

the

theorem

is

applicable

only

in

connection

with

the

experimental

situations

described

above

Even

for

such

experiments

the

result

is

only

asymptotically

valid

As

Seidenfeld

has

objected

no

indication

is

given

of

the

rate

at

which

this

concentration

occurs

-LRB-

-RRB-

There

is

also

some

circularity

in

taking

the

concentration

theorem

as

providing

a

quantitative

justification

for

that

intuitive

predilection

that

we

all

feel

for

the

uniform

distribution

-LRB-

Jaynes

p.

-RRB-

-LRB-

since

the

maximum

entropy

distribution

in

the

absence

of

constraints

is

the

uniform

distribution

-RRB-

when

the

-LRB-

clearly

intended

-RRB-

higher

probability

-LRB-

vs.

higher

multiplicity

-RRB-

construal

of

the

concentration

theorem

follows

from

the

assignment

of

equal

probability

to

each

of

the

possible

sequences

of

experimental

outcomes

The

probability

of

N

repetitions

of

an

experiment

generating

a

particular

observed

relative

frequency

distribution

is

the

sum

of

the

probabilities

of

each

of

the

sequences

of

N

experimental

outcomes

giving

rise

to

it

Assigning

equal

positive

probability

to

all

sequences

compatible

with

the

constraints

is

merely

a

sufficient

condition

for

the

higher

probability

interpretation

of

the

theorem

but

there

does

not

seem

to

be

a

simple

alternative

assignment

of

probability

consistent

with

this

interpretation

For

all

one

knows

though

there

may

be

factors

making

some

sequences

compatible

with

the

stated

constraints

more

likely

than

others

Regarding

the

ease

with

which

maximum

entropy

solutions

are

obtained

consider

the

geometry

of

a

set

of

classical

probability

functions

compatible

with

given

constraints

Let

W

-LCB-

w

w

-RCB-

denote

a

set

of

mutually

exclusive

and

exhaustive

events

The

n

n

ties

of

these

events

determine

those

of

each

member

of

the

Boolean

algebra

of

events

over

W

the

set

of

all

subsets

of

W.

Any

probability

function

on

W

can

be

unambiguously

represented

by

its

values

p

-LRB-

w

-RRB-

p

-LRB-

w

-RRB-

The

vector

of

these

values

in

turn

may

be

represented

as

a

point

n

n

in

the

space

-LSB-

-RSB-

The

set

of

all

such

points

is

the

-LRB-

n

-RRB-

dimensional

simplex

whose

n

vertices

are

the

permutations

of

-LRB-

-RRB-

For

elementary

events

the

simplex

is

a

line

segment

for

events

a

triangle

for

events

a

tetrahedron

etc.

Constraints

on

probabilities

will

determine

a

subset

of

the

full

simplex

Those

discussed

so

far

will

determine

a

convex

polytope

of

compatible

probability

functions

In

such

cases

an

arbitrary

vertex

is

probably

easier

to

calculate

than

the

maximum

entropy

element

-LRB-

which

itself

rarely

will

be

a

vertex

-RRB-

What

about

the

risk

involved

when

the

maximum

entropy

element

is

selected

from

a

set

of

distributions

one

of

which

is

regarded

-LRB-

pace

deFinetti

-RRB-

as

correct

but

unknown

Although

we

have

not

seen

it

expressed

in

print

the

feeling

is

probably

widespread

that

the

maximum

entropy

element

is

a

safe

bet

in

the

sense

that

the

expected

or

maximum

distance

from

it

to

any

other

member

of

the

-LRB-

sub

-RRB-

set

of

probability

distributions

compatible

with

given

constraints

is

either

the

minimum

possible

or

close

to

this

minimum

For

measures

of

distance

between

probability

distributions

that

are

convex

and

symmetric

this

amounts

to

the

impression

that

it

is

close

to

the

centroid

-LRB-

average

member

-RRB-

of

the

set

Except

under

special

conditions

determined

by

the

form

of

the

constraints

given

the

maximum

entropy

element

is

guaranteed

to

coincide

with

the

centroid

only

when

the

set

is

a

singleton

or

is

the

simplex

of

all

possible

distributions

over

the

events

under

consideration

-LRB-

i.e.

when

there

are

no

constraints

beyond

the

probability

axioms

-RRB-

It

is

possible

for

the

maximum

entropy

element

of

an

infinite

set

of

distributions

to

be

an

extreme

point

Consider

the

case

of

a

coin

for

which

a

confidence

interval

for

p

-LRB-

H

-RRB-

of

-LSB-

-RSB-

has

been

determined

This

may

be

regarded

as

a

constraint

on

classical

probability

functions

over

the

set

of

elementary

events

-LCB-

H

T

-RCB-

The

set

of

points

-LRB-

p

-LRB-

H

-RRB-

p

-LRB-

T

-RRB-

-RRB-

corresponding

to

this

set

of

functions

is

a

line

segment

with

endpoints

-LRB-

-RRB-

and

-LRB-

-RRB-

The

probability

function

with

maximum

entropy

corresponds

to

the

endpoint

-LRB-

-RRB-

The

function

corresponding

to

the

centroid

-LRB-

-RRB-

would

be

the

usual

point

estimate

from

such

frequencies

To

explore

in

more

detail

the

geometric

properties

of

the

maximum

entropy

element

for

a

specific

type

of

constraint

consider

a

contingency

table

with

given

marginal

probabilities

x

=p

-LRB-

A

-RRB-

=p

-LRB-

A

-RRB-

y

=p

-LRB-

B

-RRB-

and

=p

-LRB-

B

-RRB-

B

B

A

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

x

A

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

y

These

determine

four

linear

constraints

-LRB-

one

of

which

is

always

redundant

-RRB-

on

the

four

joint

probabilities

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

and

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

x

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

y

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

The

set

S

of

solutions

to

these

equations

will

be

a

convex

polytope

within

the

tetrahedron

of

all

possible

probability

functions

over

the

four

joint

events

Any

feasible

value

for

p

-LRB-

AB

-RRB-

will

determine

the

other

three

joint

probability

values

Thus

the

solution

set

has

at

most

two

vertices

it

is

either

a

single

point

or

a

line

segment

in

four

dimensions

Let

l

-LRB-

AB

-RRB-

and

u

-LRB-

AB

-RRB-

denote

respectively

the

minimum

and

maximum

feasible

values

for

p

-LRB-

AB

-RRB-

Then

l

-LRB-

AB

-RRB-

min

p

-LRB-

AB

-RRB-

max

-LRB-

x

y

-RRB-

p

C

and

u

-LRB-

AB

-RRB-

max

p

-LRB-

AB

-RRB-

min

-LRB-

x

y

-RRB-

p

C

Indexing

as

-LRB-

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

-RRB-

the

vertices

v

and

v

of

S

-LRB-

which

may

coincide

-RRB-

are

-LRB-

l

-LRB-

AB

-RRB-

x

l

-LRB-

AB

-RRB-

y

l

-LRB-

AB

-RRB-

-LRB-

-RRB-

-LRB-

x

l

-LRB-

AB

-RRB-

-RRB-

-LRB-

u

-LRB-

AB

-RRB-

x

u

-LRB-

AB

-RRB-

y

u

-LRB-

AB

-RRB-

-LRB-

-RRB-

-LRB-

x

u

-LRB-

AB

-RRB-

-RRB-

The

centroid

of

S

is

c

-LRB-

v

v

-RRB-

The

-LRB-

unique

-RRB-

maximum

entropy

joint

distribution

is

calculated

from

the

marginals

under

the

assumption

of

probabilistic

independence

between

A

and

B

-LRB-

Lewis

-RRB-

m

-LRB-

xy

x

-LRB-

-RRB-

y

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

Define

the

eccentricity

of

an

element

p

of

S

as

ecc

-LRB-

p

S

-RRB-

d

-LRB-

p

c

-RRB-

max

d

-LRB-

p

c

-RRB-

p

S

where

c

is

the

centroid

and

d

is

Euclidean

distance

ecc

-LRB-

p

S

-RRB-

will

have

a

minimum

value

of

-LRB-

when

p

c

-RRB-

and

a

maximum

value

of

-LRB-

for

a

vertex

of

S

-RRB-

For

the

marginal

constraints

given

by

the

table

p

-LRB-

B

-RRB-

p

-LRB-

B

-RRB-

p

-LRB-

A

-RRB-

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

p

-LRB-

A

-RRB-

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

the

vertices

v

and

v

the

maximum

entropy

element

m

and

the

centroid

c

are

v

v

m

c

AB

AB

AB

AB

For

this

example

ecc

-LRB-

m

S

-RRB-

which

is

somewhat

high

For

randomly

generated

pairs

-LRB-

p

-LRB-

A

-RRB-

p

-LRB-

B

-RRB-

-RRB-

the

average

value

of

ecc

-LRB-

m

S

-RRB-

was

found

to

be

The

maximum

value

was

The

expected

value

of

ecc

-LRB-

p

S

-RRB-

for

a

randomly

selected

member

p

of

a

non-singleton

S

is

So

at

least

for

this

type

of

constraint

one

can

not

expect

the

maximum

entropy

element

to

be

a

particularly

low-risk

estimate

Intervalism

A

common

motivation

for

Bayesians

to

abandon

strictness

is

discomfort

with

finding

a

rationale

for

providing

the

real

numbers

that

the

strict

theory

says

must

exist

We

have

looked

at

one

procedure

for

obtaining

these

numbers

the

maximum

entropy

procedure

and

found

ourselves

compelled

to

side

with

the

dissenters

the

case

has

not

been

made

and

will

be

difficult

to

make

In

psychological

terms

the

difficulty

may

be

seen

as

a

difficulty

of

measurement

The

reports

of

experts

the

hypothetical

choices

of

decision

makers

may

turn

out

to

be

inconsistent

But

if

we

think

of

those

reports

and

choices

as

containing

an

element

of

error

the

inconsistency

of

the

reports

may

be

taken

to

suggest

not

the

falsity

of

the

theory

but

the

accuracy

of

the

introspections

on

the

basis

of

which

the

reports

and

choices

are

made

This

is

in

fact

typical

of

the

way

in

which

our

theories

of

error

are

developed

in

many

domains

and

so

is

a

perfectly

natural

idea

to

adopt

in

connection

with

degrees

of

belief

Nevertheless

it

is

surely

reasonable

to

question

the

theory

itself

If

we

were

to

measure

light

ray

triangles

on

the

surface

of

the

earth

and

discovered

a

systematic

bias

toward

negative

errors

it

would

be

natural

to

suppose

that

the

geometry

of

light

rays

near

the

surface

of

the

earth

was

Riemannian

rather

than

Euclidean

The

situation

is

different

with

regard

to

degrees

of

belief

since

there

is

-LRB-

we

may

suppose

-RRB-

no

systematic

bias

On

the

other

hand

with

respect

to

free

choice

among

alternatives

there

is

a

bias

first

noticed

explicitly

by

C.

A.

B.

Smith

-LRB-

-RRB-

Let

a

reasonable

person

be

offered

a

Savage-type

option

How

much

will

you

pay

for

a

ticket

that

will

return

if

the

next

toss

of

this

coin

lands

heads

Now

let

the

same

person

be

offered

another

Savage-type

option

How

much

will

you

pay

for

a

ticket

that

will

return

if

the

next

toss

of

this

coin

lands

tails

The

prices

offered

by

a

reasonable

person

will

add

up

to

less

than

He

will

not

have

a

book

made

against

him

on

the

contrary

he

will

be

in

the

bookmaker

's

position

Of

course

as

the

strict

Bayesian

will

point

out

we

can

demand

that

he

make

offers

that

add

up

to

We

can

force

him

to

do

this

We

can

impose

penalties

if

he

does

n't

But

then

the

choice

of

an

offer

is

hardly

a

free

choice

is

it

The

bias

observed

by

Smith

is

that

the

offers

add

up

to

less

than

one

the

degree

to

which

the

person

believes

tails

and

the

degree

to

which

the

person

believes

heads

add

up

to

less

than

one

Shades

of

Shafer

's

belief

functions

-LRB-

See

Section

-RRB-

But

now

let

us

ask

if

it

is

appropriate

to

talk

of

degrees

at

all

Degrees

as

degrees

of

temperature

reflect

a

continuum

of

values

isomorphic

to

a

subset

of

the

real

numbers

Who

says

beliefs

are

like

this

Keynes

-LRB-

-RRB-

explicitly

denies

it

it

is

easy

-LRB-

according

to

Keynes

-RRB-

to

imagine

two

propositions

such

that

neither

is

more

probable

than

the

other

nor

are

they

equally

probable

they

are

simply

incomparable

though

each

is

comparable

to

other

propositions

No

proposition

is

more

probable

than

a

tautology

is

less

probable

than

a

contradiction

We

have

a

partial

order

of

precisely

the

sort

exploited

by

Koopman

-LRB-

-RRB-

One

instance

of

such

an

order

is

constituted

by

an

ordering

of

intervals

where

one

interval

is

less

than

another

if

every

point

in

the

first

is

less

than

any

point

in

the

second

If

neither

of

two

intervals

is

less

than

the

other

we

simply

say

that

they

are

incomparable

There

are

two

ways

in

which

two

intervals

may

be

incomparable

they

may

overlap

or

one

may

be

properly

included

in

the

other

If

the

range

of

the

probability

function

is

to

be

intervals

in

conformity

to

the

intuitions

we

are

now

considering

that

does

n't

at

all

settle

the

question

of

what

they

represent

formally

We

may

take

their

formal

representation

to

be

a

set

of

classical

point-valued

probability

functions

This

does

not

prejudge

the

issue

of

what

rational

belief

is

One

could

say

either

A

collection

of

probability

intervals

itself

represents

rational

belief

rational

belief

is

itself

an

interval

kind

of

thing

or

A

collection

of

probability

intervals

represents

a

set

of

constraints

on

rational

belief

beliefs

are

real-valued

and

rational

beliefs

satisfy

the

interval

constraints

imposed

by

probability

One

might

add

to

the

second

view

that

in

addition

these

real-valued

rational

degrees

of

belief

should

satisfy

the

probability

calculus

as

Ramsey

-LRB-

-RRB-

argued

It

is

not

clear

what

is

accomplished

by

this

It

is

not

needed

for

protection

against

actual

dutch

books

-LRB-

since

the

deductively

rational

agent

will

not

act

so

as

to

guarantee

himself

a

loss

-RRB-

and

potential

dutch

books

deductively

require

that

the

agent

commit

himself

in

advance

to

taking

any

series

of

bets

-LRB-

at

his

odds

-RRB-

that

are

offered

him

Without

that

added

condition

there

seems

to

be

little

motivation

for

adopting

the

more

complex

view

embodied

in

our

second

alternative

This

still

allows

us

to

think

of

the

intervals

as

the

envelope

of

a

set

of

probability

functions

even

though

the

individual

probability

functions

in

the

set

have

no

psychological

or

epistemological

role

We

are

merely

adopting

a

mathematically

convenient

representation

This

representation

does

embody

some

constraints

Theorem

Suppose

the

probability

of

A

is

-LSB-

p

q

-RSB-

Then

there

must

exist

a

probability

function

in

our

set

for

which

the

probability

of

A

is

p

and

one

for

which

the

probability

of

A

is

q

Then

the

probability

of

A

must

be

-LSB-

-RSB-

according

to

these

same

probability

functions

Proof

Suppose

that

A

and

B

are

logically

exclusive

and

that

the

probability

of

A

is

-LSB-

p

q

-RSB-

and

the

probability

of

B

is

-LSB-

r

s

-RSB-

Then

the

probability

of

A

v

B

is

bounded

below

by

p

r.

Let

the

probability

functions

be

p

p

p

and

p

p

can

not

assign

a

probability

to

B

of

less

than

r

else

p

would

not

be

the

function

yielding

the

minimum

value

for

B

so

p

must

assign

at

least

p

r

to

A

v

B.

the

probability

of

A

vB

is

bounded

above

by

or

q

s

since

p

which

assigns

q

to

A

can

not

assign

more

than

s

to

B

else

p

would

not

be

the

function

yielding

the

maximum

value

for

B

and

therefore

can

not

assign

more

than

q

s

to

A

v

B

We

have

said

nothing

about

the

internal

structure

of

the

set

of

probability

functions

whose

envelope

determines

the

probability

intervals

That

will

be

the

topic

of

subsequent

sections

Nor

have

we

discussed

the

question

of

updating

how

new

evidence

impinges

on

these

sets

of

probabilities

There

is

one

more

topic

that

can

appropriately

be

mentioned

here

however

According

to

one

view

of

probability

that

takes

probabilities

to

be

intervals

-LRB-

Kyburg

-RRB-

the

probability

interval

is

determined

by

statistical

knowledge

it

is

the

statistical

knowledge

that

yields

the

bounds

of

the

interval

The

general

idea

is

that

the

probability

of

a

statement

A

is

-LSB-

p

q

-RSB-

when

A

is

equivalent

to

a

statement

that

can

be

referred

to

a

reference

class

about

which

it

is

known

that

the

measure

of

the

corresponding

target

class

lies

between

p

and

q

It

is

the

statistical

knowledge

that

yields

the

interval

and

the

known

statistical

structure

convex

or

not

that

determines

the

relations

among

probability

intervals

This

may

yield

a

slightly

different

structure

in

view

of

the

role

played

by

the

conditions

of

randomness

in

the

selection

of

a

reference

class

It

remains

true

that

the

probability

of

A

is

-LSB-

-RSB-

when

the

probability

of

A

is

-LSB-

p

q

-RSB-

because

if

R

is

the

right

reference

class

for

A

it

is

also

the

right

reference

class

for

A

But

the

result

for

disjunction

must

be

weakened

since

it

may

not

be

the

case

that

A

T

and

A

v

T

will

all

be

referred

to

a

common

reference

class

Specifically

if

one

potential

reference

class

differs

from

another

only

in

that

it

provides

a

narrower

interval

-LRB-

for

example

we

know

more

about

the

behavior

of

coins

in

general

than

about

this

particular

coin

that

I

pull

from

my

pocket

-RRB-

then

it

is

to

be

preferred

In

the

case

of

A

vT

there

is

a

possible

reference

class

whose

interval

is

bounded

as

above

by

-LSB-

p

r

q

s

-RSB-

but

this

reference

class

may

be

ruled

out

by

other

considerations

It

can

be

shown

however

that

any

appropriate

reference

class

mentions

an

interval

that

includes

that

interval

or

is

included

in

it

-LRB-

Kyburg

-RRB-

Convex

Bayesianism

As

does

intervalism

convex

Bayesianism

relaxes

the

strict

Bayesian

requirement

of

a

single

numerically

determinate

probability

function

in

any

given

context

An

agent

is

allowed

a

set

S

of

probability

functions

but

the

set

must

be

convex

for

any

p

and

p

contained

in

S

and

for

any

l

S

must

also

contain

the

function

p

where

for

any

X

in

the

algebra

of

events

in

question

p

-LRB-

X

-RRB-

l?p

-LRB-

X

-RRB-

-LRB-

-RRB-

p

-LRB-

X

-RRB-

It

is

sometimes

useful

to

consider

the

probability

p

-LRB-

A

E

-RRB-

of

an

event

A

conditional

on

the

truth

of

another

event

E

-LRB-

and

nothing

else

-RRB-

where

p

-LRB-

A

E

-RRB-

p

-LRB-

A

and

E

-RRB-

p

-LRB-

E

-RRB-

The

act

of

replacing

p

with

p

where

for

all

A

in

the

algebra

p

-LRB-

A

-RRB-

p

-LRB-

A

and

E

-RRB-

p

-LRB-

E

-RRB-

is

referred

to

as

conditionalization

-LRB-

Bacchus

et

al.

-RRB-

Levi

-LRB-

-RRB-

has

shown

that

the

set

S

of

all

distributions

p

-LRB-

A

-RRB-

p

-LRB-

A

and

E

-RRB-

p

-LRB-

E

-RRB-

obtainable

from

the

members

p

of

a

convex

set

S

is

also

convex

-LRB-

Replacing

S

with

S

is

referred

to

as

convex

conditionalization

Computational

details

for

the

special

case

of

a

convex

polytope

are

discussed

by

Snow

-LRB-

-RRB-

-RRB-

Thus

with

respect

to

representing

and

updating

probabilities

convex

Bayesianism

includes

strict

Bayesianism

as

the

special

case

in

which

S

is

a

singleton

Imprecision

vs.

Indeterminacy

In

each

of

the

examples

of

the

previous

section

the

available

information

naturally

determines

a

set

of

classical

probability

functions

Levi

-LRB-

-RRB-

distinguishes

two

attitudes

toward

such

sets

Under

the

black

box

-LRB-

Good

-RRB-

interpretation

the

agent

is

committed

to

some

single

-LRB-

unknown

-RRB-

element

of

this

set

whether

he

knows

it

or

not

This

view

is

not

incompatible

with

strict

Bayesianism

Maximum

entropy

or

other

techniques

may

be

applied

to

remove

the

residual

imprecision

in

the

determination

of

the

required

probabilities

In

contrast

the

permissibility

construal

would

regard

the

probability

judgment

relative

to

such

information

irremediably

indeterminate

and

the

agent

would

be

permitted

to

use

any

of

the

probability

functions

consistent

with

the

data

Although

we

prefer

the

permissibility

to

the

black

box

interpretation

we

would

not

regard

it

as

reasonable

for

an

agent

to

select

a

single

probability

function

from

the

permissible

set

by

some

method

like

maximum

entropy

or

otherwise

since

this

would

amount

to

rejecting

probability

functions

that

are

not

ruled

out

by

the

available

information

But

the

agent

is

not

thereby

paralyzed

As

many

authors

-LRB-

including

Levi

and

going

at

least

as

far

back

as

Wald

-LRB-

-RRB-

and

Hurwicz

-LRB-

-RRB-

-RRB-

have

recognized

-LRB-

but

for

the

most

part

have

not

discussed

in

much

mathematical

detail

-RRB-

a

decision

theory

based

on

sets

of

probability

functions

is

possible

Convex

Bayesianism

and

Decision

The

subsets

of

the

probability

simplex

picked

out

by

the

types

of

data

discussed

so

far

are

convex

sets

Further

they

are

not

arbitrary

convex

sets

but

convex

polytopes

bounded

sets

of

solutions

to

systems

of

linear

inequalities

Example

The

expected

value

of

a

spin

of

a

top

with

sides

labelled

and

is

said

to

be

This

is

consistent

with

an

infinite

set

of

biases

the

solutions

to

the

pair

of

linear

equations

p

i

Si

i

p

i

Si

This

set

is

a

line

segment

in

-LSB-

-RSB-

with

endpoints

-LRB-

-RRB-

and

-LRB-

-RRB-

Example

The

set

of

probability

functions

on

the

joint

values

of

quantities

v

v

and

v

x

y

z

with

marginals

v

v

p

v

p

x

y

xy

z

z

is

a

polytope

in

-LSB-

-RSB-

with

vertices

that

can

be

determined

via

linear

programming

-LRB-

Mattheiss

et

al.

-RRB-

Any

collection

of

probability

intervals

construed

as

a

set

of

constraints

on

real-valued

probabilities

also

determines

a

convex

set

of

classical

probability

functions

-LRB-

The

set

of

intervals

may

be

inconsistent

in

which

case

the

set

is

empty

The

collection

of

intervals

may

be

such

that

some

endpoint

values

can

not

be

achieved

by

any

of

the

probability

functions

enveloped

by

the

intervals

The

intervals

may

be

tightened

to

have

this

property

by

using

linear

programming

to

find

the

minimum

and

maximum

values

of

p

compatible

with

the

orii

ginally

stated

bounds

-RRB-

Representability

of

constraints

as

a

system

of

linear

inequalities

whose

unknowns

are

the

probabilities

of

the

events

in

question

facilitates

a

generalized

maximum

expected

utility

decision

analysis

Consider

a

decision

problem

with

m

alternatives

-LCB-

a

a

-RCB-

and

n

mutually

exclusive

m

and

exhaustive

conditions

-LCB-

c

c

-RCB-

whose

probabilities

are

independent

of

the

alternative

n

chosen

-LRB-

Jeffrey

-LRB-

-RRB-

shows

that

a

problem

in

which

independence

fails

can

be

converted

to

an

equivalent

problem

in

which

it

holds

-RRB-

Let

u

denote

the

utility

of

choosing

act

a

when

i

j

i

c

is

the

case

Then

the

expected

utility

of

a

relative

to

a

classical

probability

function

p

on

j

i

-LCB-

c

c

-RCB-

is

n

n

u

p

-LRB-

c

-RRB-

i

j

j

Sj

The

domain

-LRB-

Starr

-RRB-

of

a

D

-LRB-

a

-RRB-

is

the

set

of

all

probability

functions

on

i

i

-LCB-

c

c

-RCB-

relative

to

which

the

expected

utility

of

a

is

maximum

-LRB-

or

tied

for

maximum

-RRB-

n

i

among

all

the

alternatives

Let

S

denote

the

set

of

probability

distributions

representing

the

agent

's

belief

An

action

a

is

E-admissible

relative

to

S

iff

there

exists

a

p

S

relative

to

which

it

maximizes

i

expected

utility

-LRB-

Levi

-RRB-

This

is

equivalent

to

the

condition

S?D

-LRB-

a

-RRB-

i

For

a

convex

Bayesian

S

is

a

convex

set

and

is

-LRB-

usually

-RRB-

characterizable

as

the

solution

set

of

a

system

of

linear

inequalities

with

unknowns

p

-LRB-

c

-RRB-

as

is

D

-LRB-

a

-RRB-

D

-LRB-

a

-RRB-

is

the

set

of

all

j

i

i

solutions

to

n

p

-LRB-

c

-RRB-

j

Sj

n

n

u

p

-LRB-

c

-RRB-

u

p

-LRB-

c

-RRB-

i

j

j

j

S

S

j

j

n

n

u

p

-LRB-

c

-RRB-

u

p

-LRB-

c

-RRB-

i

j

j

mj

j

S

S

j

j

If

there

exist

solutions

to

the

combined

system

consisting

of

the

inequalities

characterizing

S

and

those

characterizing

D

-LRB-

a

-RRB-

as

can

be

determined

by

linear

programming

with

an

arbitrary

i

objective

function

say

p

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

then

a

is

E-admissible

Note

that

this

method

reduces

n

i

to

strict

Bayesian

expected

utility

maximization

when

S

is

a

unit

set

Example

For

the

decision

problem

given

by

the

matrix

c

c

c

a

and

the

constraint

c

is

at

least

twice

as

likely

as

either

c

or

c

action

a

is

uniquely

E

admissible

There

exist

solutions

-LRB-

e.g.

p

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

-RRB-

to

the

system

p

-LRB-

c

-RRB-

j

Sj

p

-LRB-

c

-RRB-

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

p

-LRB-

c

-RRB-

As

can

be

determined

by

linear

programming

there

are

no

solutions

to

the

corresponding

systems

for

a

and

a

Some

authors

seem

to

be

unaware

of

the

possibility

of

such

an

analysis

In

a

passage

discussing

the

prospects

for

a

personalistic

theory

of

upper

and

lower

probability

in

a

recent

review

article

-LRB-

Genest

et

al.

p.

-RRB-

the

statement

is

made

it

is

unclear

how

the

class

-LSB-

of

strict

Bayesian

probability

functions

enveloped

by

the

upper

and

lower

bounds

-RSB-

could

be

used

at

the

end

of

the

day

A

standard

error

is

given

without

a

point

estimate

as

it

were

More

than

one

action

may

be

E-admissible

Levi

recommends

the

use

of

criteria

based

on

security

e.g.

maximin

to

select

a

single

action

from

the

E-admissible

set

An

alternative

approach

would

be

to

attempt

to

reduce

the

size

of

the

set

S

of

probability

functions

The

smaller

the

set

S

the

fewer

the

domains

D

-LRB-

a

-RRB-

with

which

it

has

nonempty

intersections

The

i

maximum

entropy

method

may

be

regarded

as

an

extreme

version

of

this

approach

Less

objectionable

is

the

method

of

Loui

-LRB-

-RRB-

who

advocates

lowering

the

value

of

the

parameter

determining

the

width

of

Kyburgian

epistemological

probabilities

until

a

value

is

reached

relative

to

which

a

single

action

is

admissible

A

method

in

the

same

spirit

for

decision

making

given

only

marginal

probabilities

is

discussed

by

Pittarelli

-LRB-

-RRB-

Belief

Functions

It

is

probably

the

case

that

more

practitioners

are

using

belief

functions

-LRB-

Shafer

-RRB-

than

are

using

convex

sets

of

classical

probability

functions

in

order

to

overcome

what

they

view

as

the

unreasonable

demands

of

strict

Bayesianism

A

belief

function

on

a

frame

of

discernment

W

of

mutually

exclusive

and

exhaustive

events

is

a

mapping

Bel

from

the

algebra

of

subsets

of

W

to

the

interval

-LSB-

-RSB-

with

the

properties

-LRB-

-RRB-

Bel

-LRB-

-RRB-

-LRB-

-RRB-

Bel

-LRB-

W

-RRB-

-LRB-

-RRB-

Bel

-LRB-

X

Y

-RRB-

Bel

-LRB-

X

-RRB-

Bel

-LRB-

Y

-RRB-

Bel

-LRB-

X

Y

-RRB-

A

classical

probability

function

-LRB-

Bayesian

belief

function

-RRB-

is

a

special

case

in

which

the

inequality

in

condition

-LRB-

-RRB-

is

replaced

by

equality

The

appeal

of

belief

functions

is

that

they

are

not

necessarily

additive

One

may

assign

belief

of

x

to

an

event

without

assigning

to

its

complement

Complete

agnosticism

is

expressed

by

the

vacuous

belief

function

for

which

Bel

-LRB-

X

-RRB-

for

any

proper

subset

X

of

W

There

is

no

way

to

do

this

with

a

single

classical

probability

function

over

W

The

traditional

solution

would

be

to

assign

the

probability

W

to

each

of

the

singleton

subsets

of

W.

-LRB-

This

is

the

principle

of

indifference

which

as

Jaynes

-LRB-

-RRB-

notes

is

a

special

case

of

the

maximum

entropy

principle

-RRB-

It

is

thus

not

possible

within

the

strict

Bayesian

framework

to

distinguish

between

the

case

in

which

one

has

good

reason

to

have

equal

positive

strengths

of

belief

in

each

of

the

possible

events

-LRB-

based

on

knowledge

of

physical

symmetries

huge

amounts

of

statistical

data

etc.

-RRB-

and

the

case

in

which

one

has

no

basis

for

any

particular

strengths

whatsoever

-LRB-

Indirectly

of

course

one

may

distinguish

these

cases

by

the

ratio

of

p

-LRB-

x

e

b

-RRB-

to

p

-LRB-

e

b

-RRB-

where

x

is

one

of

the

possible

events

e

is

the

observed

occurrence

of

an

event

in

the

same

class

and

b

is

general

background

knowledge

If

we

have

little

knowledge

of

the

historical

frequency

of

events

of

the

same

kind

as

x

then

the

observation

of

an

event

e

of

that

kind

will

have

a

larger

influence

on

our

prediction

of

a

new

occurrence

than

if

we

have

a

lot

of

such

knowledge

Thus

by

comparing

the

ratio

of

p

-LRB-

x

e

b

-RRB-

and

p

-LRB-

e

b

-RRB-

to

the

ratio

of

p

-LRB-

x

e

-RRB-

and

p

-LRB-

e

-RRB-

we

can

tell

whether

b

contains

much

information

about

the

relative

frequency

of

events

like

x.

-RRB-

Of

course

complete

ignorance

can

be

represented

by

the

entire

simplex

of

classical

probability

functions

over

W

In

fact

as

Kyburg

has

demonstrated

-LRB-

-RRB-

any

belief

function

corresponds

to

a

convex

set

S

of

classical

probability

functions

over

W

where

Bel

-LRB-

X

-RRB-

min

p

-LRB-

X

-RRB-

p

S

but

not

conversely

There

exist

perfectly

reasonable

constraints

determining

convex

sets

S

for

which

a

belief

function

can

not

be

found

satisfying

this

correspondence

The

set

of

probability

functions

over

W

-LCB-

w

w

-RCB-

satisfying

p

-LRB-

w

-RRB-

is

one

such

set

-LRB-

j

berger

et

al.

-RRB-

So

all

of

the

representational

power

of

the

belief

function

formalism

and

more

is

possessed

by

convex

Bayesianism

In

summary

Convex

Bayesianism

includes

strict

Bayesianism

as

a

special

case

Convex

Bayesianism

includes

the

theory

of

belief

functions

as

a

special

case

Conditionalization

of

a

convex

set

of

probability

functions

yields

another

convex

set

of

functions

The

convex

set

representation

allows

for

more

flexibility

and

faithfulness

to

available

information

than

does

the

strict

Bayesian

representation

In

the

usual

convex

case

the

information

determines

a

set

of

probability

functions

that

can

be

characterized

as

the

set

of

solutions

to

a

system

of

linear

inequalities

This

makes

possible

an

expected-utility

decision

methodology

based

on

linear

programming

Problems

for

convex

Bayesianism

There

are

natural

constraints

on

probability

judgments

that

can

not

be

represented

by

convex

sets

of

classical

probability

functions

These

constraints

can

however

be

captured

by

nonconvex

sets

of

functions

It

will

be

shown

that

it

is

possible

to

devise

decision

methods

for

such

sets

Disjunctive

constraints

Suppose

that

one

is

informed

that

a

particular

six-sided

die

has

been

manufactured

in

such

a

way

that

either

the

outcome

is

favored

at

the

expense

of

by

or

conversely

and

is

otherwise

fair

The

possible

biases

-LRB-

either

of

which

may

be

adopted

as

probabilities

for

the

toss

outcomes

-RRB-

are

-LCB-

-LRB-

-RRB-

-LRB-

-RRB-

-RCB-

-LRB-

-RRB-

This

is

not

a

convex

set

Neither

is

the

physically

more

realistic

-LCB-

-LRB-

e

e

-RRB-

e

-LSB-

-RSB-

-RCB-

-LCB-

-LRB-

e

e

-RRB-

e

-LSB-

-RSB-

-RCB-

which

is

the

union

of

two

disjoint

infinite

convex

sets

It

may

be

argued

that

these

numbers

represent

possible

frequencies

which

in

turn

justify

a

range

of

permissible

probabilities

with

probabilities

construed

as

betting

rates

I.e.

the

frequencies

-LRB-

-RRB-

make

acceptable

any

odds

in

the

range

to

for

a

bet

on

However

while

any

number

in

this

range

represents

reasonable

odds

for

a

single

bet

we

know

that

in

the

long

run

only

one

extreme

or

the

other

will

represent

a

break-even

set

of

odds

for

a

sequence

of

bets

In

the

long

run

any

odds

other

than

one

of

the

extremes

is

doomed

to

lead

to

loss

Note

that

since

the

biases

-LRB-

-RRB-

are

not

representable

by

a

convex

set

of

probability

functions

with

sample

space

equal

to

the

set

of

six

possible

die

toss

outcomes

they

are

not

representable

by

a

single

Dempster-Shafer

belief

function

with

this

set

as

its

frame

of

discernment

-LRB-

This

follows

from

results

in

-LRB-

Kyburg

-RRB-

-RRB-

The

closest

it

seems

possible

to

come

is

to

assign

probability

masses

m

-LRB-

-LCB-

-RCB-

-RRB-

m

-LRB-

-LCB-

-RCB-

-RRB-

m

-LRB-

-LCB-

-RCB-

-RRB-

m

-LRB-

-LCB-

-RCB-

-RRB-

m

-LRB-

-LCB-

-RCB-

-RRB-

-LRB-

Bel

-LRB-

X

-RRB-

m

-LRB-

A

-RRB-

-RRB-

But

there

seems

to

be

no

way

to

specify

that

the

mass

of

on

-LCB-

-RCB-

S

A?X

must

either

all

go

to

-LCB-

-RCB-

or

all

go

to

-LCB-

-RCB-

Independence

Judgments

of

irrelevance

-LRB-

conditional

irrelevance

-RRB-

that

is

probabilistic

independence

-LRB-

conditional

independence

-RRB-

are

often

made

are

natural

to

make

can

be

made

reliably

and

provide

computational

advantages

-LRB-

Pearl

-RRB-

An

extreme

case

illustrating

the

computational

benefits

is

that

of

n

k-valued

variables

a

probability

distribution

over

the

joint

values

of

which

is

required

Under

the

assumption

of

complete

probabilistic

independence

of

the

variables

any

of

the

joint

probabilities

is

computable

as

the

product

n

p

-LRB-

X

X

x

x

-RRB-

Pp

-LRB-

X

x

-RRB-

n

n

j

j

j

n

Estimating

the

marginal

probabilities

requires

only

kn

n

numbers

vs.

k

for

the

joint

probabilities

Even

if

it

is

the

joint

probabilities

that

are

determined

initially

the

data

may

be

stored

and

transmitted

subsequently

in

the

form

of

the

spectacularly

more

compact

marginal

probabilities

-LRB-

Lewis

-RRB-

Unfortunately

the

constraint

of

independence

combined

with

information

that

itself

would

determine

a

convex

set

of

probability

functions

will

result

in

a

nonconvex

set

The

simplest

imaginable

example

is

given

by

Richard

Jeffrey

-LRB-

-RRB-

who

points

out

that

the

set

of

probability

functions

expressing

the

irrelevance

to

each

other

of

a

given

pair

of

propositions

and

nothing

more

is

not

convex

Conditional

probabilistic

independence

-LRB-

of

which

probabilistic

independence

is

a

special

case

-RRB-

is

also

incompatible

with

a

convex

set

representation

This

is

unfortunate

since

it

is

the

basis

for

influence

diagram

-LRB-

Shachter

-RRB-

and

Bayesian

network

-LRB-

Pearl

-RRB-

techniques

which

make

computationally

feasible

the

application

of

classical

probability

to

the

representation

and

management

of

uncertainty

in

domains

of

realistic

size

Example

Suppose

that

interval-valued

probability

distributions

have

been

determined

for

the

propositional

variables

X

Y

and

Z

as

X

Y

i

Y

Z

i

XY

YZ

x

y

-LSB-

-RSB-

y

z

-LSB-

-RSB-

x

y

-LSB-

-RSB-

y

z

-LSB-

-RSB-

x

y

-LSB-

-RSB-

y

z

-LSB-

-RSB-

x

y

-LSB-

-RSB-

y

z

-LSB-

-RSB-

The

set

P

of

classical

real-valued

distributions

over

variables

X

and

Y

compatible

with

the

XY

interval

distribution

i

is

the

set

of

distributions

p

such

that

p

-LRB-

-RRB-

is

a

value

in

the

interval

XY

XY

XY

i

-LRB-

-RRB-

and

similarly

for

P

Either

set

is

convex

the

set

of

solutions

to

a

system

of

linear

XY

YZ

inequalities

The

set

P

of

all

classical

distributions

on

-LCB-

X

Y

Z

-RCB-

whose

marginals

on

XYZ

-LCB-

X

Y

-RCB-

and

-LCB-

Y

Z

-RCB-

are

elements

of

P

and

P

respectively

is

also

a

convex

set

But

the

XY

YZ

subset

I

of

classical

distributions

that

in

addition

satisfy

the

constraint

that

variables

X

and

XYZ

Z

are

conditionally

independent

given

Y

is

not

convex

The

distributions

p

and

p

XYZ

XYZ

below

are

elements

of

I

but

the

distribution

XYZ

q

p

p

XYZ

XYZ

XYZ

is

not

It

fails

to

satisfy

the

conditional

independence

constraints

q

-LRB-

xyz

-RRB-

q

-LRB-

xy

-RRB-

q

-LRB-

yz

-RRB-

q

-LRB-

y

-RRB-

XYZ

XY

YZ

Y

X

Y

Z

p

p

q

XYZ

XYZ

XYZ

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

Therefore

I

is

not

convex

XYZ

Betting

on

Independent

Events

Suppose

that

an

agent

knows

that

a

coin

is

biased

toward

tails

but

that

he

thinks

that

it

and

its

tossing

mechanism

yield

a

set

of

outcomes

with

a

binomial

distribution

In

particular

suppose

that

he

thinks

that

the

probability

of

heads

on

a

single

toss

is

in

the

interval

-LSB-

-RSB-

It

is

a

deductive

consequence

of

these

assumptions

that

the

agent

's

beliefs

regarding

the

four

possible

outcomes

of

a

pair

of

tosses

are

bounded

by

the

pairs

of

numbers

listed

below

HH

HT

TH

TT

low

high

Two

real-valued

distributions

satisfying

the

independence

and

upper

and

lower

probability

constraints

are

Toss

Toss

p

p

H

H

H

T

T

H

T

T

According

to

convex

Bayesian

doctrine

any

mixture

of

distributions

belonging

to

the

representation

of

beliefs

also

belongs

so

the

agent

should

include

in

his

set

the

mixture

q

p

p

Toss

Toss

q

H

H

H

T

T

H

T

T

Observe

that

this

set

of

betting

odds

on

the

outcome

of

a

pair

of

tosses

is

perfectly

coherent

this

is

what

we

expect

the

mixture

of

a

pair

of

coherent

betting

functions

is

coherent

But

now

in

accordance

with

the

wiley-antagonist

model

of

the

Dutch

Book

Theorem

-LRB-

Ramsey

deFinetti

-RRB-

let

the

agent

post

odds

in

accordance

with

this

coherent

distribution

of

beliefs

The

wiley

antagonist

-LRB-

a

-RRB-

sells

our

agent

a

ticket

for

that

returns

if

HH

and

nothing

otherwise

That

's

fair

Likewise

it

is

fair

for

our

antagonist

-LRB-

b

-RRB-

to

buy

from

our

agent

for

a

ticket

that

returns

if

HT

and

nothing

otherwise

Any

collection

of

such

bets

on

pairs

of

tosses

would

be

considered

fair

Let

us

say

that

an

agent

is

booked

in

expectation

if

whatever

the

true

state

of

the

world

consistent

with

his

beliefs

his

long

run

expectation

is

negative

The

expectation

of

this

set

of

bets

is

negative

almost

everywhere

on

the

interval

-LSB-

-RSB-

In

the

long

run

relative

to

his

own

beliefs

our

agent

is

almost

sure

to

lose

This

may

be

seen

as

follows

We

compute

the

expectation

of

the

pair

of

bets

-LRB-

a

-RRB-

and

-LRB-

b

-RRB-

from

the

wiley

fox

's

point

of

view

as

a

function

of

the

probability

p

of

heads

The

payoffs

for

each

outcome

are

HH

HT

TH

TT

-LRB-

a

-RRB-

-LRB-

b

-RRB-

net

The

expected

payoff

to

the

wiley

fox

is

then

p

p

This

function

is

concave

it

is

positive

for

all

values

of

p

-LSB-

-RSB-

except

the

endpoints

at

both

of

which

it

equals

reaching

a

maximum

of

for

p

The

fox

's

expectation

is

positive

almost

everywhere

regardless

of

the

value

of

p

relative

to

what

both

the

agent

and

the

fox

know

But

the

agent

's

odds

are

perfectly

coherent

Although

the

distribution

q

is

a

convex

combination

of

distributions

satisfying

the

upper

and

lower

probability

constraints

and

the

independence

constraint

it

fails

to

satisfy

the

latter

Had

the

agent

posted

odds

in

accordance

with

a

distribution

q

exhibiting

independence

the

antagonist

's

long-run

expectation

-LRB-

and

his

own

-RRB-

relative

to

q

would

have

been

zero

To

see

this

let

a

and

b

denote

the

return

value

of

the

tickets

sold

by

the

antagonist

and

by

the

agent

respectively

Let

w

q

-LRB-

HH

-RRB-

x

q

-LRB-

HT

-RRB-

y

q

-LRB-

TH

-RRB-

and

z

q

-LRB-

TT

-RRB-

The

table

of

payoffs

to

the

antagonist

is

HH

HT

TH

TT

-LRB-

a

-RRB-

a

aw

aw

aw

aw

-LRB-

b

-RRB-

bx

b

bx

bx

bx

net

a

aw

bx

b

aw

bx

aw

bx

aw

bx

The

antagonist

's

expectation

relative

to

q

is

then

w

-LRB-

a

aw

bx

-RRB-

x

-LRB-

b

aw

bx

-RRB-

y

-LRB-

aw

bx

-RRB-

z

-LRB-

aw

bx

-RRB-

which

equals

if

x

w

-LRB-

w

-RRB-

w

w

y

w

w

z

-LRB-

w

-RRB-

i.e.

if

q

embodies

the

independence

constraint

Convex

pooling

The

traditional

solution

to

the

problem

of

pooling

a

set

of

classical

probability

functions

is

to

replace

them

with

a

member

of

their

convex

hull

the

set

of

all

possible

convex

combinations

of

them

A

probability

function

p

is

a

convex

combination

of

functions

p

p

iff

n

n

there

exist

l

l

such

that

j

j

Sj

p

l

p

l

p

n

n

One

may

choose

to

work

instead

with

the

entire

convex

hull

of

the

set

of

probability

functions

under

consideration

This

is

the

approach

of

Levi

-LRB-

-RRB-

who

regards

any

of

the

elements

of

the

convex

hull

to

be

a

permissible

resolution

of

the

conflict

among

the

functions

and

hence

permissible

for

use

in

decision

making

using

the

criterion

of

E-admissibility

But

it

can

affect

the

outcome

of

a

decision

problem

if

instead

of

a

nonconvex

set

A

its

convex

hull

conv

-LRB-

A

-RRB-

is

regarded

as

the

set

of

permissible

functions

If

an

action

is

E

admissible

relative

to

a

set

A

of

probability

functions

then

it

is

E-admissible

relative

to

its

convex

hull

-LRB-

since

A?conv

-LRB-

A

-RRB-

-RRB-

but

not

conversely

Consider

the

set

of

probability

functions

A

-LCB-

p

p

-RCB-

where

p

-LRB-

s

-RRB-

p

-LRB-

s

-RRB-

p

-LRB-

s

-RRB-

and

p

-LRB-

s

-RRB-

p

-LRB-

s

-RRB-

p

-LRB-

s

-RRB-

and

the

decision

matrix

c

c

c

a

Relative

to

set

A

only

actions

a

and

a

are

-LRB-

E

-RRB-

admissible

a

is

not

eu

-LRB-

a

p

-RRB-

eu

-LRB-

a

p

-RRB-

eu

-LRB-

a

p

-RRB-

eu

-LRB-

a

p

-RRB-

Relative

to

conv

-LRB-

A

-RRB-

all

three

are

admissible

-LRB-

For

example

a

maximizes

utility

for

the

bability

function

p

p

-RRB-

Because

the

set

of

all

probability

functions

relative

to

which

an

action

maximizes

utility

is

convex

if

there

is

consensus

on

an

admissible

action

on

the

part

of

members

of

a

group

who

agree

on

utilities

but

disagree

on

probabilities

then

that

same

action

is

admissible

for

any

function

within

the

convex

hull

of

their

individual

probability

functions

When

there

is

not

consensus

it

is

not

obvious

that

a

sufficient

condition

for

admissibility

of

a

group

action

is

that

it

maximize

expected

utility

relative

to

some

convex

combination

of

the

group

probabilities

But

if

any

such

function

is

permissible

then

any

action

maximizing

expected

utility

relative

to

one

of

these

functions

should

be

admissible

This

consequence

of

the

permissibility

of

all

convex

combinations

of

permissible

probability

functions

clashes

for

example

with

the

intuition

of

Savage

-LRB-

-RRB-

that

group

minimax

is

a

reasonable

group

decision

criterion

The

group

minimax

rule

prescribes

selection

of

an

action

such

that

the

largest

loss

faced

by

any

member

of

the

group

will

be

as

small

as

possible

where

the

loss

associated

with

an

action

by

a

member

of

the

group

is

the

difference

between

its

expectation

for

that

member

and

the

expectation

of

the

action

with

maximum

expectation

for

that

member

There

exist

decision

problems

for

which

the

action

maximizing

expectation

for

a

convex

combination

of

group

probabilities

is

not

a

group

minimax

solution

although

the

action

maximizing

expectation

for

a

probability

distribution

outside

the

convex

hull

of

the

group

probabilities

is

a

group

minimax

solution

Consider

a

group

containing

three

members

facing

a

decision

among

three

actions

who

recognize

three

relevant

states

of

nature

and

agree

on

the

utility

matrix

above

Suppose

that

the

group

does

not

wish

to

consider

mixed

actions

and

that

the

group

opinions

are

c

c

c

p

Then

action

a

is

optimal

-LRB-

maximizes

expectation

-RRB-

for

members

and

and

action

a

is

optimal

for

member

For

the

mixture

of

distributions

p

p

p

p

action

a

is

optimal

However

a

is

the

group

minimax

solution

It

is

optimal

for

infinitely

many

distributions

outside

conv

-LRB-

-LCB-

p

p

p

-RCB-

-RRB-

for

example

p

-LRB-

s

-RRB-

p

-LRB-

s

-RRB-

p

-LRB-

s

-RRB-

Suppose

that

the

-LRB-

classical

-RRB-

probability

assessments

of

n

experts

are

to

be

combined

to

form

a

single

probability

function

Examples

can

be

constructed

to

demonstrate

that

if

each

expert

assesses

his

probabilities

in

a

manner

consistent

with

the

belief

in

the

probabilistic

independence

of

some

of

the

events

in

the

algebra

under

consideration

a

convex

combination

of

the

n

functions

will

not

necessarily

exhibit

the

independence

relations

agreed

on

by

the

experts

If

these

relations

are

merely

numerical

artifacts

then

the

failure

to

preserve

them

is

not

a

failure

of

the

pooling

method

On

the

other

hand

there

will

be

situations

in

which

the

experts

each

do

intend

for

certain

independence

relations

to

hold

-LRB-

and

may

actually

use

them

as

a

guide

in

the

construction

of

their

full

probability

function

over

the

algebra

of

events

-RRB-

Convex

pooling

of

probability

functions

exhibits

what

may

be

called

the

marginalization

effect

With

a

fixed

set

of

pooling

weights

the

result

of

pooling

a

set

of

probability

functions

and

then

marginalizing

is

the

same

as

the

result

of

first

marginalizing

the

functions

individually

and

then

pooling

While

some

regard

this

as

a

point

in

favor

of

convex

pooling

schemes

-LRB-

McConway

-RRB-

we

do

not

Example

Radio

station

WXYZ

has

for

many

years

used

the

weather

predictions

of

a

meteorologist

who

is

about

to

retire

but

who

has

recommended

two

younger

colleagues

Mr.

P

and

Mr.

Q

as

possible

replacements

The

station

manager

calls

them

and

asks

them

to

assess

the

probabilities

of

the

following

events

it

will

rain

tomorrow

and

the

sun

will

rise

the

day

after

it

will

rain

and

the

sun

will

not

rise

it

will

not

rain

tomorrow

and

the

sun

will

rise

the

day

after

and

it

will

not

rain

and

the

sun

will

not

rise

She

includes

the

sunrise

question

as

a

test

if

either

meteorologist

gives

positive

probability

to

the

sun

not

rising

then

she

will

not

take

seriously

the

probability

given

to

rain

by

him

-LRB-

The

dummy

events

need

only

be

events

about

which

these

alleged

experts

should

have

opinions

close

to

some

known

standard

-RRB-

Suppose

the

assessments

she

receives

are

p

Sunrise

RS

Yes

No

pR

Yes

Rainfall

No

and

q

Sunrise

RS

Yes

No

qR

Yes

Rainfall

No

Given

p

and

q

and

a

convex

pooling

scheme

the

choice

of

weights

should

be

and

RS

RS

But

it

would

make

sense

for

them

to

be

something

like

and

given

only

p

and

q

R

R

However

even

for

weights

of

and

the

marginal

distribution

that

the

station

manager

takes

as

representing

the

weighted

pooled

opinions

of

the

two

p

remains

disturbingly

close

to

q

R

R

on

any

reasonable

measure

of

the

closeness

of

probability

distributions

More

information

is

available

to

the

pooler

in

the

original

probability

functions

than

is

available

in

the

marginals

-LRB-

Lindley

-RRB-

since

the

marginals

are

deducible

from

the

original

functions

but

usually

not

conversely

Whether

as

in

the

example

the

information

bears

on

the

reliability

of

the

experts

or

not

there

are

bound

to

be

situations

in

which

the

extra

information

is

wasted

if

one

adopts

a

pooling

method

exhibiting

the

marginalization

effect

Finally

consider

a

disreputable

traveling

circus

one

of

whose

practices

is

to

have

a

clown

sell

bags

of

peanuts

some

of

which

it

is

announced

to

the

crowd

contain

coupons

redeemable

for

a

colorful

helium-inflated

balloon

Suppose

that

detectives

with

binoculars

and

two-way

radios

are

in

attendance

at

one

of

the

performances

Every

ten

seconds

in

a

prearranged

sequence

a

different

detective

is

to

radio

his

code

name

and

his

current

pointvalued

estimate

of

the

probability

that

the

next

bag

selected

will

contain

a

coupon

to

a

van

parked

outside

the

circus

tent

The

sequence

of

estimates

received

up

to

the

time

that

the

reports

mysteriously

stop

is

The

sounds

of

peanut

sales

continue

to

be

received

in

the

van

for

several

minutes

after

the

final

report

is

received

It

would

be

more

reasonable

to

choose

an

estimate

in

the

range

-LRB-

-RRB-

by

extrapolation

of

some

sort

than

to

choose

an

estimate

in

the

range

-LSB-

-RSB-

say

as

would

result

from

a

convex

pooling

scheme

It

may

be

objected

that

the

van

occupant

and

the

radio

station

manager

in

the

two

previous

examples

may

be

treated

as

experts

themselves

To

adopt

their

probability

assessment

-LRB-

whatever

it

turns

out

to

be

-RRB-

may

not

be

to

adopt

a

probability

assessment

in

the

convex

hull

of

the

expert

assessments

canvassed

by

them

but

it

is

to

adopt

an

element

of

-LRB-

extreme

point

of

-RRB-

the

convex

hull

of

the

set

consisting

of

their

own

assessment

plus

the

solicited

assessments

Besides

being

dictatorial

this

strikes

us

as

not

the

type

of

aggregation

of

expert

opinion

that

the

proponents

of

convex

pooling

have

in

mind

Expert

opinions

are

typically

solicited

by

decision

makers

who

are

not

themselves

experts

in

the

domains

of

the

decision

problems

they

face

In

these

two

examples

the

station

manager

is

not

a

weather

forecaster

and

it

may

be

that

the

van

occupant

has

poor

vision

Set-based

Bayesianism

The

representational

inadequacies

of

the

convex

set

characterization

of

uncertainty

may

be

overcome

by

the

use

of

sets

of

classical

probability

functions

that

are

not

necessarily

convex

but

at

some

computational

cost

for

decision

making

Before

examining

the

technical

problems

for

a

more

general

set-based

Bayesian

decision

theory

we

first

examine

some

related

methods

Previous

work

Gardenfors

and

Sahlin

-LRB-

-RRB-

propose

two

stages

in

the

identification

of

a

subset

of

the

probability

simplex

First

the

set

of

functions

that

do

not

contradict

the

decision

maker

's

knowledge

in

the

given

decision

situation

are

identified

as

epistemically

possible

A

realvalued

measure

of

the

epistemic

reliability

of

the

functions

in

the

epistemically

possible

set

is

introduced

that

is

in

the

spirit

of

Keynes

weight

of

evidence

Only

those

probability

functions

with

epistemic

reliability

values

above

some

threshold

are

retained

for

use

in

decision

making

Example

In

the

case

of

a

contingency

table

with

known

marginals

B

B

A

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

x

A

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

y

the

epistemically

possible

set

of

joint

probability

functions

is

E

-LCB-

p

max

-LRB-

x

y

-RRB-

p

-LRB-

AB

-RRB-

min

-LRB-

x

y

-RRB-

p

-LRB-

AB

-RRB-

x

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

y

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

p

-LRB-

AB

-RRB-

-RCB-

Suppose

that

the

identities

of

attributes

A

and

B

are

kept

secret

The

agent

may

have

extensive

experience

with

attributes

in

general

and

may

believe

that

attributes

tend

to

be

independent

It

is

reasonable

then

for

him

to

choose

a

reliability

measure

that

gives

maximum

reliability

to

and

is

symmetric

around

the

probability

function

computed

under

the

assumption

of

probabilistic

independence

of

A

and

B

The

epistemically

reliable

set

is

then

a

subset

of

E

for

which

xy

e

p

-LRB-

AB

-RRB-

xy

e

A

related

representation

of

beliefs

is

proposed

by

Nau

-LRB-

-RRB-

In

place

of

an

unrestricted

epistemic

reliability

measure

he

utilizes

a

concave

piecewise-linear

function

on

the

unit

interval

to

characterize

the

confidence

weight

associated

with

any

possible

probability

value

for

an

event

The

collection

of

these

functions

for

each

of

the

events

under

consideration

determines

a

convex

subset

of

the

probability

simplex

over

the

events

Gardenfors

and

Sahlin

endorse

selecting

the

action

whose

minimum

expected

utility

for

any

p

in

the

epistemically

reliable

set

is

maximum

This

criterion

may

result

in

selection

of

actions

that

are

not

E-admissible

Example

Let

S

-LCB-

p

p

-LRB-

c

-RRB-

-RCB-

for

the

decision

problem

c

c

a

Suppose

that

each

p

S

is

sufficiently

reliable

Action

a

has

the

maximum

minimum

expected

utility

as

p

ranges

over

S

but

there

is

no

single

p

S

relative

to

which

a

has

the

maximum

expected

utility

of

the

three

The

epistemically

reliable

sets

in

these

examples

happen

to

be

convex

Gardenfors

and

Sahlin

give

an

example

in

which

the

epistemically

reliable

set

is

not

convex

However

they

do

not

address

the

computational

problems

that

may

arise

in

attempting

to

determine

the

admissibility

of

actions

relative

to

such

sets

Determining

E-admissibility

Although

we

have

advocated

the

use

of

linear

programming

algorithms

to

determine

E

admissibility

of

actions

relative

to

linearly

constrained

sets

of

classical

probability

functions

-LRB-

which

are

adequate

to

characterize

all

of

the

convex

constraints

discussed

so

far

-RRB-

all

that

it

is

necessary

to

determine

is

whether

or

not

there

exist

any

feasible

solutions

to

the

linear

programming

problem

It

is

not

necessary

to

identify

an

optimal

solution

-LRB-

The

objective

function

can

be

just

p

-LRB-

c

-RRB-

where

c

is

one

of

the

states

for

the

decision

problem

-RRB-

Thus

for

nonj

j

linear

or

non-convex

constraints

the

fact

that

a

particular

optimization

method

is

unable

to

guarantee

that

the

local

optimum

it

identifies

will

be

a

global

optimum

does

not

mean

that

E

admissibility

can

not

be

determined

As

an

example

of

a

nonlinear

but

convex

constraint

on

probabilities

consider

a

probabilk

ity

function

p

-LCB-

X

X

-RCB-

fi

-LSB-

-RSB-

where

p

-LRB-

X

-RRB-

x

k

and

x

is

taken

to

lie

in

the

interval

-LSB-

l

u

-RSB-

X

may

be

the

event

k

successes

in

k

trials

for

some

binomial

experiment

An

action

a

is

E

i

admissible

relative

to

such

information

if

and

only

if

there

exists

an

x

-LSB-

l

u

-RSB-

for

which

k

k

k

k

x

u

-LRB-

a

X

-RRB-

-LRB-

-RRB-

u

-LRB-

a

X

-RRB-

x

u

-LRB-

a

X

-RRB-

-LRB-

-RRB-

u

-LRB-

a

X

-RRB-

i

i

j

j

for

all

j.

Example

T

has

just

met

B

who

claims

that

she

lives

with

her

teenage

daughter

T

would

like

to

ask

B

out

on

a

date

but

would

prefer

to

date

an

honest

person

He

estimates

that

the

probability

that

a

teenage

girl

will

answer

a

phone

call

placed

in

the

evening

to

a

home

occupied

only

by

that

girl

and

her

mother

is

in

the

interval

-LSB-

-RSB-

He

decides

to

call

B

's

home

phone

on

five

consecutive

evenings

see

who

answers

and

immediately

hang

up

If

B

answers

on

all

five

occasions

he

will

then

decide

to

ask

her

out

based

on

the

utilities

Honest

Dishonest

Date

No

date

The

domains

of

the

actions

are

D

-LRB-

Date

-RRB-

-LCB-

p

p

-LRB-

Honest

-RRB-

-RCB-

D

-LRB-

No

date

-RRB-

-LCB-

p

p

-LRB-

Honest

-RRB-

-RCB-

p

-LRB-

Honest

-RRB-

x

where

x

-LSB-

-RSB-

Thus

p

-LRB-

Honest

-RRB-

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

Therefore

among

other

things

T

should

not

ask

B

out

The

following

example

utilizes

the

coin

of

Section

regarding

which

an

agent

assumes

probabilistic

independence

of

toss

outcomes

and

for

which

the

agent

has

bounds

on

probabilities

of

individual

toss

outcomes

The

probabilistic

independence

constraint

introduces

nonconvexity

Example

A

coin

is

to

be

tossed

twice

Three

outcomes

are

distinguished

both

heads

both

tails

or

one

heads

and

one

tails

Let

p

-LRB-

H

-RRB-

-LSB-

-RSB-

Then

p

-LRB-

HH

-RRB-

x

p

-LRB-

HT

or

TH

-RRB-

-LRB-

-RRB-

p

-LRB-

TT

-RRB-

-LRB-

-RRB-

where

x

-LSB-

-RSB-

The

set

of

triples

S

-LCB-

-LRB-

x

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

x

-LSB-

-RSB-

-RCB-

is

not

convex

the

triples

-LRB-

-RRB-

and

-LRB-

-RRB-

are

members

of

S

but

-LRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

-LRB-

-RRB-

is

not

-LRB-

-RRB-

Suppose

that

a

decision

is

to

be

made

with

an

outcome

contingent

on

the

result

of

the

two

tosses

with

utility

matrix

HH

HT

or

TH

TT

a

For

this

example

determining

E-admissibility

involves

identifying

roots

of

quadratic

equations

Action

a

has

expected

utility

at

least

as

great

as

action

a

when

x

-LRB-

-RRB-

-LRB-

-RRB-

x

x

-LRB-

-RRB-

-LRB-

-RRB-

i.e.

when

x

i.e.

when

x

or

x

But

-LRB-

-LSB-

-RSB-

-LSB-

-RSB-

-RRB-

-LSB-

-RSB-

So

a

is

not

E-admissible

From

above

a

has

expected

utility

at

least

as

great

as

a

for

every

value

of

the

parameter

x.

a

's

expected

utility

equals

or

exceeds

a

's

when

x

x

-LRB-

-RRB-

-LRB-

-RRB-

x

x

-LRB-

-RRB-

-LRB-

-RRB-

i.e.

when

x

or

x

Since

-LRB-

-LSB-

-RSB-

-LRB-

-LSB-

-RSB-

-LSB-

-RSB-

-RRB-

-RRB-

-LSB-

-RSB-

-LSB-

-RSB-

a

is

E-admissible

Similarly

a

is

E-admissible

Decomposition

A

set

S

of

classical

probability

functions

for

a

decision

problem

may

be

not

only

nonconvex

but

nonconnected

i.e.

the

union

of

two

or

more

separated

sets

-LRB-

Lay

-RRB-

Disjunctive

constraints

on

probabilities

as

discussed

in

Section

are

characterized

by

such

sets

These

sets

are

not

representable

by

a

single

system

of

simultaneous

inequalities

However

if

an

action

is

E-admissible

relative

to

one

of

the

connected

sets

into

which

S

is

decomposable

then

it

is

E-admissible

relative

to

S

-LRB-

and

conversely

-RRB-

Recall

that

a

decision

problem

partitions

the

probability

simplex

into

convex

domains

one

for

each

action

An

action

is

E

admissible

if

and

only

if

the

intersection

of

S

and

its

domain

is

nonempty

For

S

S

S

S

D

if

and

only

if

for

some

i

S

D

-RRB-

n

i

For

a

finite

set

S

of

probability

functions

an

action

is

tested

against

each

element

individually

in

some

arbitrary

order

until

a

probability

function

relative

to

which

the

action

maximizes

expected

utility

is

found

or

the

set

is

exhausted

In

the

worst

case

the

computation

required

to

identify

the

admissible

actions

is

increased

by

a

factor

of

S

over

a

strict

Bayesian

analysis

-LRB-

not

counting

the

effort

involved

in

settling

on

a

unique

probability

function

-RRB-

For

a

finite

union

of

connected

sets

some

of

which

may

be

infinite

the

actions

are

tested

for

E

admissibility

relative

to

each

set

A

decision

problem

involving

the

probabilities

of

the

biased

die

discussed

in

Section

would

be

analyzed

in

this

way

An

action

would

be

E-admissible

iff

it

were

E-admissible

relative

either

to

-LCB-

-LRB-

e

e

-RRB-

e

-LSB-

-RSB-

-RCB-

or

to

-LCB-

-LRB-

e

e

-RRB-

e

-LSB-

-RSB-

-RCB-

Some

sets

of

classical

probability

functions

are

neither

connected

nor

finite

unions

of

connected

sets

of

probability

functions

e.g.

-LCB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RCB-

but

such

sets

are

unlikely

to

be

encountered

Example

The

nonconvex

example

of

Gardenfors

and

Sahlin

-LRB-

Section

-RRB-

may

be

analyzed

by

decomposition

An

agent

is

informed

that

one

or

the

other

of

A

or

not-A

is

overwhelmingly

likely

but

not

which

one

Let

x

R

stand

for

the

function

assigning

probability

x

to

A

and

to

not-A

Gardenfors

and

Sahlin

depict

the

epistemic

reliability

measure

r

as

symmetric

about

x

with

r

-LRB-

-RRB-

and

r

-LRB-

-RRB-

r

-LRB-

-RRB-

For

definiteness

suppose

r

-LRB-

x

-RRB-

-LRB-

-RRB-

and

that

the

reliability

threshold

is

Then

the

epistemically

reliable

set

is

S

-LCB-

p

p

-LRB-

A

-RRB-

-RCB-

-LCB-

p

p

-LRB-

A

-RRB-

-RCB-

Any

decision

relative

to

the

nonconvex

set

S

thus

may

be

decomposed

into

two

decision

problems

involving

convex

sets

-LRB-

For

the

criterion

of

Gardenfors

and

Sahlin

the

overall

minimum

expected

utility

for

each

action

may

be

computed

as

the

minimum

of

the

minima

relative

to

each

of

the

convex

sets

An

action

whose

aggregate

minimum

is

maximum

is

selected

This

action

is

not

always

E-admissible

-RRB-

Conclusion

Strict

Bayesianism

is

too

strict

It

is

unreasonable

to

demand

that

an

agent

adopt

a

single

real-valued

probability

function

in

any

and

all

decision-making

contexts

Probabilities

should

be

based

on

statistical

knowledge

i.e.

knowledge

of

frequencies

the

data

usually

warrant

only

a

more-or-less

narrow

interval

of

probability

for

each

event

under

consideration

Even

if

one

somehow

has

real-valued

probabilities

for

certain

events

they

may

not

be

those

directly

relevant

for

a

particular

decision

problem

For

example

one

may

have

strict

Bayesian

marginal

or

conditional

probabilities

from

which

one

may

infer

a

convex

set

of

strict

Bayesian

probability

functions

from

which

in

turn

bounds

on

the

probabilities

of

interest

may

be

inferred

Convex

Bayesianism

on

the

other

hand

is

not

strict

enough

By

allowing

any

convex

combination

of

strict

Bayesian

probability

functions

compatible

with

the

available

information

into

the

set

of

permissible

functions

incompatible

probability

functions

may

be

introduced

This

is

most

serious

in

the

case

of

-LRB-

conditional

-RRB-

probabilistic

independence

Independence

assumptions

are

necessary

in

domains

of

realistic

size

for

efficient

management

of

uncertainty

Furthermore

such

assumptions

may

often

reliably

be

based

on

fundamental

knowledge

about

causality

Unfortunately

independence

introduces

nonconvexity

Replacing

a

not-necessarily

convex

set

of

classical

probability

functions

with

its

convex

hull

may

result

in

unnecessary

indecisiveness

an

action

may

be

E-admissible

with

respect

to

the

convex

hull

of

a

set

S

but

not

relative

to

S

itself

It

may

also

lead

to

financial

ruin

As

discussed

in

Section

posting

odds

in

accordance

with

a

probability

function

not

in

S

but

in

the

convex

hull

of

S

may

result

in

negative

long-run

expectation

relative

to

the

knowledge

embodied

in

S

A

set-based

Bayesian

will

represent

uncertainty

by

a

set

of

classical

probability

functions

as

will

the

convex

Bayesian

However

given

the

same

data

hers

will

be

a

subset

of

the

convex

Bayesian

's

representation

-LRB-

Hence

her

representation

will

be

more

informative

in

a

Dalkey-Jeffrey

sense

-RRB-

Determination

of

whether

or

not

an

action

is

E-admissible

relative

to

a

convex

or

nonconvex

set

S

for

which

a

system

of

constraints

can

be

formulated

-LRB-

we

have

discussed

examples

only

of

linear

constraints

for

the

convex

case

-RRB-

requires

determination

of

the

existence

of

a

feasible

solution

to

a

mathematical

programming

problem

it

is

not

necessary

to

identify

an

optimal

solution

Thus

it

is

not

clear

that

the

computational

costs

associated

with

a

more

flexible

set-based

Bayesianism

are

so

much

greater

that

one

should

have

to

settle

for

a

strictly

convex

alternative

to

strict

Bayesianism

ACKNOWLEDGMENTS

This

work

was

supported

in

part

by

the

National

Science

Foundation

the

Signals

Warfare

Center

of

the

U.

S.

Army

and

the

Research

Foundation

of

the

State

University

of

New

York

REFERENCES

Bacchus

F.

Kyburg

H.

and

Thalos

M.

Against

Conditionalization

Synthese

Cheeseman

P.

A

Method

of

Computing

Generalized

Bayesian

Probability

Values

for

Expert

Systems

Proc

Int

Joint

Conference

of

Artificial

Intelligence

Dalkey

N.

Inductive

Inference

and

the

Representation

of

Uncertainty

Proc

Workshop

on

Uncertainty

in

Artificial

Intelligence

deFinetti

B.

Foresight

Its

Logical

Laws

its

Subjective

Sources

in

Kyburg

and

Smokler

-LRB-

eds

-RRB-

Studies

in

Subjective

Probability

Wiley

deFinetti

B.

Theory

of

Probability

Wiley

Gardenfors

P.

and

Sahlin

N.-E.

Unreliable

Probabilities

Risk

Taking

and

Decision

Making

Synthese

Genest

C.

and

Zidek

J.

Combining

Probability

Distributions

A

Critique

and

an

Annotated

Bibliography

Statistical

Science

Good

I.

J.

Subjective

Probability

as

the

Measure

of

a

Non-measurable

Set

in

Nagel

Suppes

and

Tarski

-LRB-

eds

-RRB-

Logic

Methodology

and

Philosophy

of

Science

Stanford

Univ.

Press

Hurwicz

L.

Some

Specification

Problems

and

Applications

to

Econometric

Models

-LRB-

abstract

-RRB-

Econometrica

Jaynes

E.

T.

Prior

Probabilities

IEEE

Trans

SMC

Jaynes

E.

T.

On

the

Rationale

of

Maximum-entropy

Methods

Proc

IEEE

Jeffrey

R.

Savage

's

Omelet

in

Suppe

and

Asquith

-LRB-

eds

-RRB-

PSA

vol

Philosophy

of

Science

Association

Jeffrey

R.

Bayesianism

with

a

Human

Face

in

Earman

-LRB-

ed

-RRB-

Testing

Scientific

Theories

Univ.

of

Minnesota

Press

Jeffrey

R.

Indefinite

Probability

Judgment

A

Reply

to

Levi

Philosophy

of

Science

Jeffreys

H.

Theory

of

Probability

Oxford

Keynes

J.

M.

A

Treatise

on

Probability

Macmillan

Koopman

B.

O.

The

Bases

of

Probability

Bulletin

of

the

American

Mathematical

Soc.

Kyburg

H.

Probability

and

the

Logic

of

Rational

Belief

Wesleyan

University

Press

Middletown

CT.

Kyburg

H.

The

Reference

Class

Philosophy

of

Science

Kyburg

H.

Epistemology

and

Inference

Univ.

of

Minnesota

Press

Kyburg

H.

Bayesian

and

Non-Bayesian

Evidential

Updating

Artificial

Intelligence

Lay

S.

Convex

Sets

and

Their

Applications

Wiley

Levi

I.

On

Indeterminate

Probabilities

Journal

of

Philosophy

Levi

I.

The

Enterprise

of

Knowledge

MIT

Press

Levi

I.

Imprecision

and

Indeterminacy

in

Probability

Judgment

Philosophy

of

Science

Lewis

P.

Approximating

Probability

Distributions

to

Reduce

Storage

Requirements

Information

and

Control

Lindley

D.V.

Reconciliation

of

Discrete

Probability

Distributions

in

Bernardo

et

al.

-LRB-

eds

-RRB-

Bayesian

Statistics

North-Holland

Loui

R.

Decisions

with

Indeterminate

Probabilities

Theory

and

Decision

Macqueen

J.

and

Marschak

J.

Partial

Knowledge

Entropy

and

Estimation

Proc

Nat

Acad

of

Sciences

Mattheiss

T.

and

Rubin

D.

A

Survey

and

Comparison

of

Methods

for

Finding

all

Vertices

of

Convex

Polyhedral

Sets

Mathematics

of

Operations

Res.

McConway

K.

Marginalization

and

Linear

Opinion

Pools

J.

of

the

Am

Stat

Assoc.

Nau

R.

Decision

Analysis

with

Indeterminate

or

Incoherent

Probabilities

Annals

of

Operations

Res.

Pearl

J.

Probabilistic

Reasoning

in

Intelligent

Systems

Networks

of

Plausible

Inference

Morgan

Kaufmann

Pittarelli

M.

Decisions

with

Probabilities

over

Finite

Product

Spaces

to

appear

in

IEEE

Trans

SMC

Ramsey

F.

P.

Truth

and

probability

in

Braithwaite

-LRB-

ed

-RRB-

The

Foundations

of

Mathematics

and

Other

Logical

Essays

Routledge

and

Kegan

Paul

Savage

L.

J.

The

Foundations

of

Statistics

Dover

Seidenfeld

T.

The

Behavioral

and

Brain

Sciences

Seidenfeld

T.

Entropy

and

Uncertainty

Philosophy

of

Science

Shachter

R.

Probabilistic

Inference

and

Influence

Diagrams

Operations

Res.

Shafer

G.

A

Mathematical

Theory

of

Evidence

Princeton

Univ.

Press

Smith

C.

A.

B.

Personal

Probabilities

and

Statistical

Analysis

J.

Royal

Stat

Soc

Ser

A.

Snow

P.

Bayesian

Inference

without

Point

Estimates

Proc

AAAI

Snow

P.

Improved

Posterior

Probability

Estimates

from

Prior

and

Conditional

Linear

Constraint

Systems

IEEE

Trans

SMC

Starr

M.

A

Discussion

of

Some

Normative

Criteria

for

Decision-making

under

Uncertainty

Industrial

Mgmt

Rev.

Stirling

W.

and

Morrell

D.

Convex

Bayes

Decision

Theory

IEEE

Trans

SMC

Wald

A.

Statistical

Decision

Functions

Wiley

Weichselberger

K.

and

Pohlman

S.

A

Methodology

for

Uncertainty

in

KnowledgeBased

Systems

Springer-Verlag

Zwick

R.

and

Wallsten

T.

Combining

Stochastic

Uncertainty

and

Linguistic

Inexactness

Theory

and

Experimental

Evaluation

of

Four

Fuzzy

Probability

Models

in

Gaines

and

Boose

-LRB-

eds

-RRB-

Machine

Learning

and

Uncertain

Reasoning

Academic

Press