2
Belief, probability and exchangeability
We ﬁrst discuss what properties a reasonable belief function should have, and
show that probabilities have these properties. Then, we review the basic ma
chinery of discrete and continuous random variables and probability distribu
tions. Finally, we explore the link between independence and exchangeability.
2.1 Belief functions and probabilities
At the beginning of the last chapter we claimed that probabilities are a way
to numerically express rational beliefs. We do not prove this claim here (see
Chapter 2 of Jaynes (2003) or Chapters 2 and 3 of Savage (1972) for details),
but we do show that several properties we would want our numerical beliefs
to have are also properties of probabilities.
Belief functions
Let
F
,
G
, and
H
be three possibly overlapping statements about the world.
For example:
F
=
{
a person votes for a leftofcenter candidate
}
G
=
{
a person’s income is in the lowest 10% of the population
}
H
=
{
a person lives in a large city
}
Let Be() be a
belief function
, that is, a function that assigns numbers to
statements such that the larger the number, the higher the degree of belief.
Some philosophers have tried to make this more concrete by relating beliefs
to preferences over bets:
•
Be(
F
)
>
Be(
G
) means we would prefer to bet
F
is true than
G
is true.
We also want Be() to describe our beliefs under certain conditions:
•
Be(
F

H
)
>
Be(
G

H
) means that if we knew that
H
were true, then we
would prefer to bet that
F
is also true than bet
G
is also true.
P.D. Hoﬀ,
A First Course in Bayesian Statistical Methods
,
Springer Texts in Statistics, DOI 10.1007/9780387924076
2,
c
±
Springer Science+Business Media, LLC 2009
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2 Belief, probability and exchangeability
•
Be(
F

G
)
>
Be(
F

H
) means that if we were forced to bet on
F
, we would
prefer to do it under the condition that
G
is true rather than
H
is true.
Axioms of beliefs
It has been argued by many that any function that is to numerically represent
our beliefs should have the following properties:
B1
Be(not
H

H
)
≤
Be(
F

H
)
≤
Be(
H

H
)
B2
Be(
F
or
G

H
)
≥
max
{
Be(
F

H
)
,
Be(
G

H
)
}
B3
Be(
F
and
G

H
) can be derived from Be(
G

H
) and Be(
F

G
and
H
)
How should we interpret these properties? Are they reasonable?
B1
says that the number we assign to Be(
F

H
), our conditional belief in
F
given
H
, is bounded below and above by the numbers we assign to complete
disbelief (Be(not
H

H
)) and complete belief (Be(
H

H
)).
B2
says that our belief that the truth lies in a given set of possibilities should
not decrease as we add to the set of possibilities.
B3
is a bit trickier. To see why it makes sense, imagine you have to decide
whether or not
F
and
G
are true, knowing that
H
is true. You could do this
by ﬁrst deciding whether or not
G
is true given
H
, and if so, then deciding
whether or not
F
is true given
G
and
H
.
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 Spring '10
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 Probability, Probability theory, Yi

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