Bayesian Decision Theory

1
Bayesian Decision Theory

• Foundations for a unified theory

2
What is it?

• 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.

3

• 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).

4
Plan

• Introductory remarks
• Prospects
• Basic Bayesian hypotheses
• Representation theorems
• A short history
• Ramseys 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 (2nd lecture)

5
Types of prospects

• 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)

6
Main Claims

• 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 XY, respectively weighted by the conditional
  probability, given X, of XY and of XY.
• Adams Thesis The rational degree of belief to
  have in P?X is the conditional probability of X
  given that P.

7
Representation Theorems

• 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.

8
Ramsey-Savage Framework

• Worlds / consequences ?1, ?2, ?3,
• Propositions / events P, Q, R,
• Conditional Prospects / Actions (P??1)(Q??2),

• 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

9
Ramseys Solution to the Measurement Problem

• Ethically neutral propositions
• Problem of definition
• Enp P has probability one-half iff for all ?1 and
  ?2
• (P??1)(P??2) ? (P??1)(P??2)
• Differences in value
• Values are sets of equi-preferred prospects
• ? - ß ? ? d iff (P??)(P?d) ? (P? ß )(P??)

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• Existence of utility
• Axiomatic characterisation of a value difference
  structure implies that existence of a mapping
  from values to real numbers such that
• - ß ? d iff U(?) U(ß) U(?) U(d)
• Derivation of probability
• Suppose d ? (? if P)(ß if P). Then

11
Evaluation

• The Justification problem
• Why should measurement axioms hold?
• Sure-Thing Principle versus P4 and Impartiality
• Jeffreys objection
• Fanciful causal hypotheses and artifacts of
  attribution.
• Behaviourism in decision theory
• Ethical neutrality versus state dependence
• Desirabilistic dependence
• Constant acts

12
Utility Dependence
13
Probability Dependence
14
Jeffrey

• 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

15
Conditionals

• Two types of conditional?
• Counterfactual If Oswald hadnt killed Kennedy
  then someone else would have.
• Indicative If Oswald didnt 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

16
Lewis-Stalnaker semantics

• 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
  AB-world there is a closer AB-world (relative to
  an ordering on worlds).
• Limit assumption There is a closest world
• Uniqueness Assumption There is at most one
  closest world.

17
The Ramsey-Adams Hypothesis

• 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)
• 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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• A Common Logic
• AB ? A?B ? A?B
• ??A ? A
• A?A ? ?
• A?A ? ?
• A?B ? A?AB
• (A?B)(A?C) ? A?BC
• (A?B) v (A?C) ? A?(B v C)
• (A?B) ? A?B

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The Bombshell

• 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 non-trivial assignment of
  truth-conditions to the conditional consistent
  with the Ramsey-Adams hypothesis.
• Conclusion
• few philosophical theses that have been more
  decisively refuted Joyce (1999, p.191)
• Ditch bivalence!

20
Boolean algebra
?
A?B
B?C
A?C
C
B
A
?
21
Conditional Algebras (1)
?
A?C?A?C
A?B
B?C
A?C
A?C?A
A?C?C
C
B
A
A?C?AC
(X?Y)(X?Z) ? X?YZ (X?Y) v (X?Z) ? X?(Y v Z)
?
22
Conditional algebras (2)
?
A?C?A?C
A?B
B?C
A?C
A?C?A
A?C?C
C
B
A
A?C?AC
XY ? X?Y
?
23
Conditional algebras (3)
?
A?C?A?C
A?B
B?C
A?C
A?C?A
A?C?C
C
B
A
A?C?AC
X?Y ? X?Y
?
24
Normally bounded algebras (1)
?
A?C?A?C
A?B
B?C
A?C
A?C?A
A?C?C
C
B
A
A?C?AC
X?X ? ? X?Y ? X?XY
?
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Material Conditional
?
A?C?A?C
A?B
B?C
A?C
A?C?A
A?C?C
C
B
A
A?C?AC
X?? ? X
?
26
Normally bounded algebras (2)
?
A?C?A?C
A?B
B?C
A?C
A?C?A
A?C?C
C
B
A
A?C?AC
X?X ? ? (X?Y) ? X?Y
?
27
Conditional algebras (3)
?
A?B
B?C
A?C
A?C?A
A?C?C
C
B
A
?