1 bayesian decision theory foundations for a unified theory
TRANSCRIPT
![Page 1: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/1.jpg)
1
Bayesian Decision Theory
Foundations for a unified theory
![Page 2: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/2.jpg)
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.
![Page 3: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/3.jpg)
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).
![Page 4: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/4.jpg)
4
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)
![Page 5: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/5.jpg)
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)
![Page 6: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/6.jpg)
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 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.
![Page 7: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/7.jpg)
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.
![Page 8: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/8.jpg)
8
Ramsey-Savage Framework1. Worlds / consequences: ω1, ω2, ω3, …
2. Propositions / events: P, Q, R, …
3. Conditional Prospects / Actions: (P→ω1)(Q→ω2), …
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 …“
![Page 9: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/9.jpg)
9
Ramsey’s Solution to the Measurement Problem
1. Ethically neutral propositions• Problem of definition
• Enp P has probability one-half iff for all ω1 and ω2
(P→ω1)(¬P→ω2) (¬P→ω1)(P→ω2)
2. Differences in value• Values are sets of equi-preferred prospects - β γ – δ iff (P→)(¬P→δ) (P→ β )(¬P→γ)
![Page 10: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/10.jpg)
10
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(
UU
UUP
![Page 11: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/11.jpg)
11
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
![Page 12: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/12.jpg)
12
Utility Dependence
Good egg Rotten egg
Break egg 6-egg omelette
None wasted
Nothing to eat
5 eggs wasted
Throw egg away 5-egg omelette
1 egg wasted
5-egg omelette
None wasted
Good egg Rotten egg
Miracle 6-egg omelette
None wasted
6-egg omelette
None wasted
Topsy Turvy Nothing to eat
5 eggs wasted
6-egg omelette
None wasted
![Page 13: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/13.jpg)
13
Probability Dependence
Republican Democrat
Dodgy land deal Low taxes
Unrestricted development
High taxes
Restricted development
No deal No development No development
Miracle deal High taxes
Restricted development
Low taxes
Unrestricted development
![Page 14: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/14.jpg)
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
![Page 15: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/15.jpg)
15
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]
![Page 16: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/16.jpg)
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 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.
![Page 17: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/17.jpg)
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.
)|()( ABpBAp
![Page 18: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/18.jpg)
18
A Common Logic
1. AB AB AB
2. A A
3. AA 4. A¬A 5. AB AAB
6. (AB)(AC) ABC
7. (AB) v (AC) A(B v C)
8. ¬(AB) A¬B
![Page 19: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/19.jpg)
19
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: 1. “few philosophical theses that have been more decisively
refuted” – Joyce (1999, p.191)
2. Ditch bivalence!
![Page 20: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/20.jpg)
20
A CB
AB BCAC
Boolean algebra
![Page 21: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/21.jpg)
21
A CB
AB BCAC
ACA ACC
ACAC
ACAC
Conditional Algebras (1)
(XY)(XZ) XYZ(XY) v (XZ) X(Y v Z)
![Page 22: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/22.jpg)
22
A CB
AB BCAC
ACA ACC
ACAC
ACAC
Conditional algebras (2)
XY XY
![Page 23: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/23.jpg)
23
A CB
AB BCAC
ACA ACC
ACAC
ACAC
Conditional algebras (3)
XY XY
![Page 24: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/24.jpg)
24
A CB
AB BCAC
ACA ACC
ACAC
ACAC
Normally bounded algebras (1)
XX XY XXY
![Page 25: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/25.jpg)
25
A CB
AB BCAC
ACA ACC
ACAC
ACAC
Material Conditional
X ¬X
![Page 26: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/26.jpg)
26
A CB
AB BCAC
ACA ACC
ACAC
ACAC
Normally bounded algebras (2)
X¬X ¬(XY) X¬Y
![Page 27: 1 Bayesian Decision Theory Foundations for a unified theory](https://reader036.vdocuments.us/reader036/viewer/2022062517/56649f2e5503460f94c481d6/html5/thumbnails/27.jpg)
27
A CB
AB BCAC
ACA ACC
Conditional algebras (3)