Bayesian Decision Theory

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# Bayesian Decision Theory - PowerPoint PPT Presentation

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

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…
• 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.
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).

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
• A common logic
• Conditional algebras
• A Unified Theory (2nd lecture)
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
• 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
• 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
• 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
• 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)
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
• 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
• 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
• 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

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.
• 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.
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
• Question: What must the truth-conditions of AB be, in order that Ramsey-Adams hypothesis be satisfied?

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

AB

AC

BC

B

C

A

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)

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)

ACAC

AB

AC

BC

ACA

ACC

B

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ACAC

XYXY

Normally bounded algebras (1)

ACAC

AB

AC

BC

ACA

ACC

B

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A

ACAC

• XX 
• XY XXY

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)

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 (3)

AB

AC

BC

ACA

ACC

B

C

A