1

• Bayesian decision theories are formal models of rational agency, typically comprising a theory of:
– Consistency of belief, desire and preference
– Optimal choice
• Lots of common ground…
– Ontology : Agents; states of the world; actions/options; consequences
– Form : Two variable quantitative models ; centrality of representation theorem
– Content : The principle that rational action maximises expected benefit.
2

It seems natural therefore to speak of plain Decision
Theory . But there are differences too ...
e.g. Savage versus Jeffrey.
– Structure of the set of prospects
– The representation of actions
– SEU versus CEU.
Are they offering rival theories or different expressions of the same theory?
Thesis : Ramsey, Savage, Jeffrey (and others) are all special cases of a single Bayesian Decision Theory
(obtained by restriction of the domain of prospects).
3

• Introductory remarks
– Prospects
– Basic Bayesian hypotheses
– Representation theorems
• A short history
– Ramsey’s solution to the measurement problem
– Ramsey versus Savage
– Jeffrey
• Conditionals
– Lewis-Stalnaker semantics
– The Ramsey-Adams Hypothesis
– A common logic
• Conditional algebras
• A Unified Theory (2 nd lecture)
4

• Usual factual possibilities e.g. it will rain tomorrow; UK inflation is 3%; etc.
– Denoted by P, Q, etc.
– Assumed to be closed under Boolean compounding
• Conjunction: PQ
• Negation: ¬P
• Disjunction: P v Q
• Logical truth/falsehood:
T ,

• Plus derived conditional possibilities e.g. If it rains tomorrow our trip will be cancelled; if the war in Iraq continues, inflation will rise.
– The prospect of X if P and Y if Q will be represented as
(P→X)(Q→Y)
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• Probability Hypothesis : Rational degrees of belief in factual possibilities are probabilities.
• SEU Hypothesis : The desirability of (P→X)(¬P→Y) is an average of the desirabilities of PX and ¬PY, respectively weighted by the probability that P or that ¬P.
• CEU Hypothesis : The desirability of the prospect of X is an average of the desirabilities of XY and X¬Y, respectively weighted by the conditional probability, given X, of XY and of X¬Y.
• Adams Thesis : The rational degree of belief to have in
P→X is the conditional probability of X given that P.
6

• Two problems; one kind of solution!
– Problem of measurement
– Problem of justification
• Scientific application : Representation theorems shows that specific conditions on (revealed) preferences suffice to determine a measure of belief and desire.
• Normative application : Theorems show that commitment to conditions on (rational) preference imply commitment to properties of rational belief and desire.
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1. Worlds / consequences: ω
1
, ω
2
, ω
3
, …
2.
Propositions / events: P, Q, R, …
3.
Conditional Prospects / Actions: (P→ω
1
)(Q→ω
2
), …
Good egg Rotten egg
Break egg 6-egg omelette Nothing to eat
Throw egg away 5-egg omelette 5-egg omelette
4. Preferences are over worlds and conditional prospects.
“If we had the power of the almighty … we could by offering him options discover how he placed them in order of merit
…“
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1. Ethically neutral propositions
• Problem of definition
• Enp P has probability one-half iff for all ω
1 and ω
2
(P→ω
1
)(¬P→ω
2
)
 (¬P→ω
1
)(P→ω
2
)
2. Differences in value
• Values are sets of equi-preferred prospects
• 
β  γ – δ iff (P→ 
) (¬P→δ)  (P→ β )(¬P→γ)
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3. Existence of utility
Axiomatic characterisation of a value difference structure implies that existence of a mapping from values to real numbers such that:

β = γ – δ iff U( 
) – U(β) = U(γ) –U(δ)
4. Derivation of probability
Suppose δ 
(
 if P)( β if ¬P). Then:
Pr( P )

U (

)
U (

)

U (

)

U (

)
10

• The Justification problem
– Why should measurement axioms hold?
– Sure-Thing Principle versus P4 and Impartiality
• Jeffrey’s objection
– Fanciful causal hypotheses and artifacts of attribution.
– Behaviourism in decision theory
• Ethical neutrality versus state dependence
– Desirabilistic dependence
– Constant acts
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Break egg
Throw egg away
Miracle
Topsy Turvy
Good egg
6-egg omelette
None wasted
5-egg omelette
1 egg wasted
Good egg
6-egg omelette
None wasted
Nothing to eat
5 eggs wasted
Rotten egg
Nothing to eat
5 eggs wasted
5-egg omelette
None wasted
Rotten egg
6-egg omelette
None wasted
6-egg omelette
None wasted
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Republican
Dodgy land deal Low taxes
Unrestricted development
Democrat
High taxes
Restricted development
No deal No development No development
Miracle deal High taxes
Restricted development
Low taxes
Unrestricted development
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• Advantages
– A simple ontology of propositions
– State dependent utility
– Partition independence (CEU)
• Measurement
– Under-determination of quantitative representations
– The inseparability of belief and desire?
– Solutions: More axioms, more relations or more prospects?
• The logical status of conditionals
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• Two types of conditional?
– Counterfactual : If Oswald hadn’t killed Kennedy then someone else would have.
– Indicative : If Oswald didn’t kill Kennedy then someone else did
Two types of supposition
– Evidential : If its true that …
– Interventional : If I make it true that …
[Lewis, Joyce, Pearl versus Stalnaker, Adams,
Edgington]
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Intuitive idea : A □ →B is true iff B is true in those worlds most like the actual one in which A is true.
Formally : A □ →B is true at a world w iff for every A¬B world there is a closer AB -world (relative to an ordering on worlds).
1. Limit assumption : There is a closest world
2. Uniqueness Assumption : There is at most one closest world.
16

• General Idea : Rational belief in conditionals goes by conditional belief for their consequents on the assumption that their antecedent is true.
• Adams Thesis : The probability of an (indicative) conditional is the conditional probability of its consequent given its antecedent:
(AT) p ( A

B )
 p ( B | A )
• Logic from belief : A sentence Y can be validly inferred from a set of premises iff the high probability of the premises guarantees the high probability of Y .
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1. AB  A

B  A

B
2.
 
A

A
3.
A

A
 
4.
A
 ¬ A
 
5. A

B

A

AB
6. (A

B)(A

C)

A

BC
7. (A

B) v (A

C)

A

(B v C)
8.
¬ (A

B)

A
 ¬B
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• Question : What must the truth-conditions of A

B be, in order that Ramsey-Adams hypothesis be satisfied?
• Answer : The question cannot be answered.
Lewis, Edgington, Hajek, Gärdenfors, Döring , …: There is no nontrivial assignment of truth-conditions to the conditional consistent with the Ramsey-Adams hypothesis.
• Conclusion :
1.
“few philosophical theses that have been more decisively refuted” – Joyce (1999, p.191)
2.
Ditch bivalence!
19

A

B

A

C
B

C
A
B C

20

A

B
A

A

C
B

A

C

A

C
B

C
A

C

A A

C

C
C
A

C

AC
(X

Y)(X

Z)

X

YZ
(X

Y) v (X

Z)

X

(Y v Z)
21

A

B
A

A

C
B

C
A

C

A

C
B

C
A

C

A A

C

C
A

C

AC
XY  X

Y
22

A

B
A

A

C
B

C
A

C

A

C
B

C
A

C

A A

C

C
A

C

AC
X

Y  X

Y
23

A

B
A

A

C

A

C
A

C B

C
A

C

A A

C

C
B C

A

C

AC
X

X
 
X

Y

X

XY
24

A

B
A

A

C
B

C
B

C
A

C

A

C
A

C

A A

C

C
A

C

AC
X
   ¬X
25

A

B
A

A

C

A

C
A

C B

C
A

C

A A

C

C
B C

A

C

AC
X
 ¬X  
¬(X 
Y)

X
 ¬Y
26


A

B
A

C

A
A
A

C
B
B

C
A

C

C
C

27