1. Probability Disassembled
John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

3. How should we use axiom systems for the probability calculus?

4. Prix Fixe

5. À la carte

6. First English edition of Euclid’s Elements

7. Theorems of Euclid’s geometry. Euclid’s postulates
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5ONE. Through any given point can be drawn exactly one straight line parallel to a given line.
Euclid’s postulates
5NONE. Through any given point NO straight lines can be drawn parallel to a given line.
Theorems of
spherical geometry.
5MORE. Through any given point MORE than one straight line can be drawn parallel to a given line.
Theorems of
hyperbolic geometry.
Theorems of
Euclid’s geometry.

8. An axiom system tailored to be used à la carte
Framework
Addition
Bayes Property
• Narrowness
• “Rescale and refute”
Real Values
Independent
and qualitative

9. Framework

11. Universal Comparability
[A|B] < [C|D] or [A|B] = [C|D] or [A|B] > [C|D]
Fails for interval valued representations.
0.1
0.5
0.4
0.6
0.9 1
This atom of Radium 221 decayed within 30 seconds
Radium 221 has a half life of 30 seconds. [ | ]
Fails for very different outcome spaces.
versus

12. Addition

13. Underlying intuition
The range of degrees of confirmation span justification of complete belief and complete disbelief.
Support for A
Support for not-A
Ignorance is precluded.
Bel(A) = Bel(not-A) = 0
Bel(A or not-A) = 1

14. Bayes Property

15. Underlying intuition H E H E P(H&E|E) = P(H|E) Narrowness
An hypothesis H accrues inductive support from evidence E just if it has a disjunctive part that entails the evidence. H E
Narrowness
The presence of other disjunctive parts logically incompatible with the evidence does not affect the level of support. H E
P(H&E|E) = P(H|E)
‘Refute and Rescale’…

16. Specific Conditioning Logic[ ]
> [ ] | |
Canary
or whale
Canary
Bird
Bird

17. Underlying intuition = ‘Refute and Rescale’
Evidence bears on hypotheses H1, H2 that entail it by
• refuting those logically incompatible with it H3 and
• uniformly redistributing support over those that remain;
this uniform redistribution is carried out everywhere in the same way and preserves the relative ranking of hypotheses that entail the evidence. H3 H1 H2 E
• Inductive content always provided by priors.
• No Bayesian computation is inductively complete.
P(H1|E) P(E|H1) x P(H1) P(H1)
P(H2|E) P(E|H2) x P(H2) P(H2) = 1
Elsewhere: All calculi of inductive inference, Bayesian or otherwise, are incomplete.

18. Real Values

19. Strengths are… … Real valued. Interval valued. Multidimensional.
So far, all properties are qualitative.
Real valued.
Interval valued.
Multidimensional.
Real and infinitesimal valued. …
Just leave as qualitative since sometimes support is imprecise and cannot be captured quantitatively.