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 • The Ramsey-Adams Hypothesis • 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 • 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 • 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.
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.
A Common Logic • ABABAB • A A • AA • A¬A • AB AAB • (AB)(AC) ABC • (AB) v (AC) A(B v C) • ¬(AB)A¬B
The Bombshell • Question: What must the truth-conditions of AB 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 AB AC BC B C A
Conditional Algebras (1) ACAC AB AC BC ACA ACC B C A ACAC • (XY)(XZ) XYZ • (XY) v (XZ) X(Y v Z)
Conditional algebras (2) ACAC AB AC BC ACA ACC B C A ACAC XY XY
Conditional algebras (3) ACAC AB AC BC ACA ACC B C A ACAC XYXY
Normally bounded algebras (1) ACAC AB AC BC ACA ACC B C A ACAC • XX • XY XXY
Material Conditional ACAC AB AC BC ACA ACC B C A ACAC X ¬X
Normally bounded algebras (2) ACAC AB AC BC ACA ACC B C A ACAC • X¬X • ¬(XY) X¬Y
Conditional algebras (3) AB AC BC ACA ACC B C A