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Bayesian Decision Theory. Foundations for a unified theory. 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…

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bayesian decision theory

Bayesian Decision Theory

Foundations for a unified theory

what is it
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.
slide3
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).

slide4
Plan
  • 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 (2nd lecture)
types of prospects
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)
main claims
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 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.
representation theorems
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.
ramsey savage framework
Ramsey-Savage Framework
  • Worlds / consequences: ω1, ω2, ω3, …
  • Propositions / events: P, Q, R, …
  • Conditional Prospects / Actions: (P→ω1)(Q→ω2), …
  • 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 …“
ramsey s solution to the measurement problem
Ramsey’s Solution to the Measurement Problem
  • Ethically neutral propositions
    • Problem of definition
    • Enp P has probability one-half iff for all ω1 andω2

(P→ω1)(¬P→ω2)  (¬P→ω1)(P→ω2)

  • Differences in value
    • Values are sets of equi-preferred prospects
    •  - β  γ – δ iff (P→)(¬P→δ)  (P→ β )(¬P→γ)
slide10
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)

Derivation of probability

Suppose δ ( if P)(β if ¬P). Then:

evaluation
Evaluation
  • 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
jeffrey
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
conditionals
Conditionals
  • 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]

lewis stalnaker semantics
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 A¬B-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.
the ramsey adams hypothesis
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.
slide18
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
the bombshell
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!
boolean algebra
Boolean algebra

AB

AC

BC

B

C

A

conditional algebras 1
Conditional Algebras (1)

ACAC

AB

AC

BC

ACA

ACC

B

C

A

ACAC

  • (XY)(XZ) XYZ
  • (XY) v (XZ) X(Y v Z)

conditional algebras 2
Conditional algebras (2)

ACAC

AB

AC

BC

ACA

ACC

B

C

A

ACAC

XY XY

conditional algebras 3
Conditional algebras (3)

ACAC

AB

AC

BC

ACA

ACC

B

C

A

ACAC

XYXY

normally bounded algebras 1
Normally bounded algebras (1)

ACAC

AB

AC

BC

ACA

ACC

B

C

A

ACAC

  • XX 
  • XY XXY

material conditional
Material Conditional

ACAC

AB

AC

BC

ACA

ACC

B

C

A

ACAC

X ¬X

normally bounded algebras 2
Normally bounded algebras (2)

ACAC

AB

AC

BC

ACA

ACC

B

C

A

ACAC

  • X¬X 
  • ¬(XY) X¬Y

conditional algebras 327
Conditional algebras (3)

AB

AC

BC

ACA

ACC

B

C

A