## Philosophies of probability: objective Bayesianism and its challenges (2004)

Venue: | Handbook of the philosophy of mathematics. Elsevier, Amsterdam. Handbook of the Philosophy of Science |

Citations: | 9 - 5 self |

### BibTeX

@INPROCEEDINGS{Williamson04philosophiesof,

author = {Jon Williamson},

title = {Philosophies of probability: objective Bayesianism and its challenges},

booktitle = {Handbook of the philosophy of mathematics. Elsevier, Amsterdam. Handbook of the Philosophy of Science},

year = {2004},

publisher = {Elsevier}

}

### OpenURL

### Abstract

This chapter presents an overview of the major interpretations of probability followed by an outline of the objective Bayesian interpretation and a discussion of the key challenges it faces.

### Citations

7314 |
Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference
- Pearl
- 1988
(Show Context)
Citation Context ... to the counterintuitive conclusion. Keynes himself had stressed the importance of taking qualitative knowledge into account and the difficulties that ensue if qualitative information is ignored: 48 (=-=Pearl, 1988-=-, p. 468; Hunter, 1989) 49 (Hunter, 1989, p. 91) 21sBernoulli’s second axiom, that in reckoning a probability we must take everything into account, is easily forgotten in these cases of statistical pr... |

6503 |
A mathematical theory of communication
- Shannon
- 1948
(Show Context)
Citation Context ...uffered shipwreck; which one is guessed to be the one that was destroyed? If I considered merely the number of ships, I 29 (Jaynes, 1988) 30 (Bernoulli, 1713) 31 (Laplace, 1814) 32 (Keynes, 1921) 33 (=-=Shannon, 1948-=-) 34 (Jaynes, 1957) 35 (Rosenkrantz, 1977; Jaynes, 2003) 13swould conclude that the misfortune could have happened to each of them with equal chance; but because I remember that one of them had been e... |

3125 | Convergence of Probability Measures - Billingsley - 1968 |

1081 |
The logic of scientific discovery
- Popper
- 2002
(Show Context)
Citation Context ... in London in 2005, then V determines a repeatable experiment, namely the selection of vehicles at random in London in 2005, and thus there is a natural propensity interpretation. Suppose, on the 15 (=-=Popper, 1934-=-, Chapter VIII) 16 (Popper, 1959; Popper, 1983, Part II) 17 Popper (1983, pp. 290 and 355). It is important to stress that the axioms of this section and the last had a different status for Popper tha... |

865 | Biological sequence analysis: probabilistic models of proteins and nucleic acids
- Durbin, Eddy, et al.
- 1998
(Show Context)
Citation Context ... Lewis (1994, p. 479) just plays the optimism card: ‘if nature is kind to us, the problem needn’t arise.’ 60 (Lewis, 1994, p. 482) 61 (Manning and Schütze, 1999) 62 (Quaife, 1992; Schumann, 2001) 63 (=-=Durbin et al., 1999-=-) 28suncertain knowledge of deterministic chemical reactions in cell metabolism. In a probability logic, or progic for short, probability is combined with logic in one or more of the following two way... |

783 |
Foundations of statistical natural language processing
- Manning, Schütze
- 2002
(Show Context)
Citation Context ... noisy relational data to forecasting toxicity from 59In response Lewis (1994, p. 479) just plays the optimism card: ‘if nature is kind to us, the problem needn’t arise.’ 60 (Lewis, 1994, p. 482) 61 (=-=Manning and Schütze, 1999-=-) 62 (Quaife, 1992; Schumann, 2001) 63 (Durbin et al., 1999) 28suncertain knowledge of deterministic chemical reactions in cell metabolism. In a probability logic, or progic for short, probability is ... |

706 |
Information Theory and Statistical Mechanics
- Jaynes
(Show Context)
Citation Context ...which one is guessed to be the one that was destroyed? If I considered merely the number of ships, I 29 (Jaynes, 1988) 30 (Bernoulli, 1713) 31 (Laplace, 1814) 32 (Keynes, 1921) 33 (Shannon, 1948) 34 (=-=Jaynes, 1957-=-) 35 (Rosenkrantz, 1977; Jaynes, 2003) 13swould conclude that the misfortune could have happened to each of them with equal chance; but because I remember that one of them had been eaten away by rot a... |

564 | Inducing features of random fields - Pietra, Pietra, et al. - 1997 |

553 | Probability theory: The logic of science
- Jaynes
- 2003
(Show Context)
Citation Context ...at was destroyed? If I considered merely the number of ships, I 29 (Jaynes, 1988) 30 (Bernoulli, 1713) 31 (Laplace, 1814) 32 (Keynes, 1921) 33 (Shannon, 1948) 34 (Jaynes, 1957) 35 (Rosenkrantz, 1977; =-=Jaynes, 2003-=-) 13swould conclude that the misfortune could have happened to each of them with equal chance; but because I remember that one of them had been eaten away by rot and old age more than the others, had ... |

383 | A Treatise on Probability
- Keynes
- 1921
(Show Context)
Citation Context ...that one of them suffered shipwreck; which one is guessed to be the one that was destroyed? If I considered merely the number of ships, I 29 (Jaynes, 1988) 30 (Bernoulli, 1713) 31 (Laplace, 1814) 32 (=-=Keynes, 1921-=-) 33 (Shannon, 1948) 34 (Jaynes, 1957) 35 (Rosenkrantz, 1977; Jaynes, 2003) 13swould conclude that the misfortune could have happened to each of them with equal chance; but because I remember that one... |

280 |
An Unsolvable Problem of Elementary Number Theory
- Church
- 1936
(Show Context)
Citation Context ...Gambling systems are ineffective: if Vf is determined by a recursive place selection f, then for each v, freq Vf (v) = freq V(v). 11 (Venn, 1866) 12 (Reichenbach, 1935) 13 (von Mises, 1928, 1964) 14 (=-=Church, 1936-=-) 7sGiven a collective V we can then define—following von Mises—the probability of v to be the frequency of v in V: p(v) =df freq V(v). Clearly freq V(v) ≥ 0. Moreover � v@V |v|n � V = n so v@V freq n... |

274 | An analysis of first-order logics of probability - Halpern - 1990 |

247 | Scientific reasoning: The Bayesian approach (2nd edition - Howson, Urbach - 1993 |

242 | Truth and probability - Ramsey - 1926 |

230 |
Reasoning about Uncertainty
- Halpern
- 2003
(Show Context)
Citation Context ...′(ψ) = 1}) > 1/2. Consequence is defined straightforwardly in 64 Williamson (2002) presents a more comprehensive survey. 65 (Howson, 2001, 143) 66 (Howson, 2001, Theorem 1) 67 (Howson, 2001, 150) 68 (=-=Halpern, 2003-=-, §7.3) 30sterms of satisfiability by worlds. Halpern later extends the above propositional language to a first-order language and introduces frequency terms ||ψ|| X , interpreted as ‘the frequency wi... |

220 | A Logic for Reasoning about Probabilities - Fagin, Halpern, et al. - 1990 |

174 | Prior Probabilities - Jaynes - 1968 |

174 | The selection of prior distributions by formal rules - Kass, Wasserman - 1996 |

149 |
Foundations of the theory of probability
- Kolmogorov
- 1950
(Show Context)
Citation Context ... probability functions are usually thought of as extensions of p and denoted by the same letter p. Thus p can be construed as a function that maps each u@U ⊆ V to a non-negative real number. p can 1 (=-=Kolmogorov, 1933-=-) 3sbe further extended to assign numbers to conjunctions tu of assignments where t@T ⊆ V, u@U ⊆ V : if t ∼ u then tu is an assignment to T ∪ U and p(tu) is the marginal probability awarded to tu@(T ∪... |

143 |
Where do we stand on maximum entropy
- Jaynes
- 1979
(Show Context)
Citation Context ...py principle ran like this: given that degrees of belief ought to be maximally non-committal, Shannon’s information theory shows us that they are entropy-maximising probabilities. 39 This type of 38 (=-=Jaynes, 1979-=-, pp. 40–41 of the original 1978 lecture) 39 (Jaynes, 1957) 15sjustification assumes from the outset that some kind of logical norm is desired. On the other hand, axiomatic derivations of the maximum ... |

127 |
The emergence of probability
- Hacking
- 1975
(Show Context)
Citation Context ...a single-case variable does not vary. The value of a single-case variable may not be known, however, and one can still think of the variable as taking a range of possible values. 8 (Gillies, 2000) 9 (=-=Hacking, 1975-=-) 10 Warning: some authors, such as Popper (1983, §3.3) and Gillies (2000, p. 20), use the term ‘objective’ for what I call ‘physical’. However, their terminology has the awkward consequence that the ... |

103 |
A Subjectivist’s Guide to Objective Chance
- Lewis
- 1980
(Show Context)
Citation Context ...han simply a small sample frequency—plenty of large samples of relevant events, and plenty of relevant qualitative information, for instance.) Lewis instead takes 56 See Williamson (2005a, §5.3). 57 (=-=Lewis, 1980-=-, 1994) 58 (Lewis, 1994, p. 475) 27schances to be products of the best system of laws, the best way of systematising the universe. The problem is that the criteria for comparing systems of laws—a bala... |

95 |
A Philosophical Essay on Probabilities
- Laplace
- 1951
(Show Context)
Citation Context ...me it is announced that one of them suffered shipwreck; which one is guessed to be the one that was destroyed? If I considered merely the number of ships, I 29 (Jaynes, 1988) 30 (Bernoulli, 1713) 31 (=-=Laplace, 1814-=-) 32 (Keynes, 1921) 33 (Shannon, 1948) 34 (Jaynes, 1957) 35 (Rosenkrantz, 1977; Jaynes, 2003) 13swould conclude that the misfortune could have happened to each of them with equal chance; but because I... |

94 | The Continuum of Inductive Methods - Carnap - 1952 |

88 | Bayes or Bust - Earman - 1992 |

71 | Foresight: Its logical laws, its subjective sources - Finetti - 1937 |

67 |
2000), Philosophical Theories of Probability
- Gillies
(Show Context)
Citation Context ...ause the value of a single-case variable does not vary. The value of a single-case variable may not be known, however, and one can still think of the variable as taking a range of possible values. 8 (=-=Gillies, 2000-=-) 9 (Hacking, 1975) 10 Warning: some authors, such as Popper (1983, §3.3) and Gillies (2000, p. 20), use the term ‘objective’ for what I call ‘physical’. However, their terminology has the awkward con... |

62 |
The Uncertain Reasoner’s Companion
- Paris
- 1992
(Show Context)
Citation Context ...al to non-committal objective priors rather than embark on a laborious process of introspection, elicitation or analysis of sensitivity of posterior to choice of prior. 40 (Paris and Vencovská, 1990; =-=Paris, 1994-=-; Paris and Vencovská, 2001) 16sA third motivating argument appeals to caution. In many applications of probability the risks attached to bold predictions that turn out wrong are high. For instance, a... |

53 | Bayesian nets and causality: philosophical and computational foundations - Williamson - 2005 |

52 |
A note on the inevitability of maximum entropy
- Paris, Vencovská
- 1990
(Show Context)
Citation Context ...ns now (often tacitly) appeal to non-committal objective priors rather than embark on a laborious process of introspection, elicitation or analysis of sensitivity of posterior to choice of prior. 40 (=-=Paris and Vencovská, 1990-=-; Paris, 1994; Paris and Vencovská, 2001) 16sA third motivating argument appeals to caution. In many applications of probability the risks attached to bold predictions that turn out wrong are high. Fo... |

50 |
Concerning measures in first order calculi
- Gaifman
- 1964
(Show Context)
Citation Context ...uired in applications of an external progic. To reiterate, given some premiss sentences 69 (Halpern, 2003, §10.3) 70 (Paris, 1994, Theorem 7.9; Paris and Vencovská, 2001) 71 (Paris, 1994, Chapter 11; =-=Gaifman, 1964-=-) 72 (Paris, 1994, Chapter 12) 31sθ1, . . . , θn and their probabilities x1, . . . , xn we often want to know what probability y to give to a conclusion sentence φ of interest—not to churn out all φ y... |

49 |
Realism and the Aim of Science
- Popper
- 1982
(Show Context)
Citation Context ...able experiment, namely the selection of vehicles at random in London in 2005, and thus there is a natural propensity interpretation. Suppose, on the 15 (Popper, 1934, Chapter VIII) 16 (Popper, 1959; =-=Popper, 1983-=-, Part II) 17 Popper (1983, pp. 290 and 355). It is important to stress that the axioms of this section and the last had a different status for Popper than they did for von Mises. Von Mises used the f... |

45 |
Automated Development of Fundamental Mathematical Theories
- Quaife
- 1990
(Show Context)
Citation Context ...asting toxicity from 59In response Lewis (1994, p. 479) just plays the optimism card: ‘if nature is kind to us, the problem needn’t arise.’ 60 (Lewis, 1994, p. 482) 61 (Manning and Schütze, 1999) 62 (=-=Quaife, 1992-=-; Schumann, 2001) 63 (Durbin et al., 1999) 28suncertain knowledge of deterministic chemical reactions in cell metabolism. In a probability logic, or progic for short, probability is combined with logi... |

40 | Decidability and expressiveness for first-order logics of probability - Abadi, Halpern - 1994 |

38 | Probability, Statistics and Truth - Mises - 1957 |

37 | Mathematical Theory of Probability and Statistics - Mises - 1964 |

34 | Objective Bayesian methods for model selection: introduction and comparison - Berger, Pericchi - 2001 |

31 |
Humean Supervenience Debugged
- Lewis
- 1994
(Show Context)
Citation Context ...le frequency—plenty of large samples of relevant events, and plenty of relevant qualitative information, for instance.) Lewis instead takes 56 See Williamson (2005a, §5.3). 57 (Lewis, 1980, 1994) 58 (=-=Lewis, 1994-=-, p. 475) 27schances to be products of the best system of laws, the best way of systematising the universe. The problem is that the criteria for comparing systems of laws—a balance between simplicity ... |

26 | The well-posed problem - Jaynes - 1973 |

26 | Automated Theorem Proving in Software Engineering
- Schumann
- 2001
(Show Context)
Citation Context ...y from 59In response Lewis (1994, p. 479) just plays the optimism card: ‘if nature is kind to us, the problem needn’t arise.’ 60 (Lewis, 1994, p. 482) 61 (Manning and Schütze, 1999) 62 (Quaife, 1992; =-=Schumann, 2001-=-) 63 (Durbin et al., 1999) 28suncertain knowledge of deterministic chemical reactions in cell metabolism. In a probability logic, or progic for short, probability is combined with logic in one or more... |

24 |
The propensity interpretation of probability
- Popper
- 1959
(Show Context)
Citation Context ...mines a repeatable experiment, namely the selection of vehicles at random in London in 2005, and thus there is a natural propensity interpretation. Suppose, on the 15 (Popper, 1934, Chapter VIII) 16 (=-=Popper, 1959-=-; Popper, 1983, Part II) 17 Popper (1983, pp. 290 and 355). It is important to stress that the axioms of this section and the last had a different status for Popper than they did for von Mises. Von Mi... |

22 |
Probabilities over rich languages
- Gaifman, Snir
- 1982
(Show Context)
Citation Context ..., the problem here is that there is no reason to suppose that agents will give probability 0 to the same assignments. One might try to provide such 25 (de Finetti, 1937; Gillies, 2000, pp. 69–83) 26 (=-=Gaifman and Snir, 1982-=-, §2) 11sa guarantee by bolstering subjective Bayesianism with a rationality constraint that says that agents must be undogmatic, i.e., they must only give probability 0 to logically impossible assign... |

22 |
Method and Decision: Towards a Bayesian Philosophy of Science
- Rosenkrantz, Inference
- 1977
(Show Context)
Citation Context ...ed to be the one that was destroyed? If I considered merely the number of ships, I 29 (Jaynes, 1988) 30 (Bernoulli, 1713) 31 (Laplace, 1814) 32 (Keynes, 1921) 33 (Shannon, 1948) 34 (Jaynes, 1957) 35 (=-=Rosenkrantz, 1977-=-; Jaynes, 2003) 13swould conclude that the misfortune could have happened to each of them with equal chance; but because I remember that one of them had been eaten away by rot and old age more than th... |

20 | Representation dependence in probabilistic inference
- Halpern, Koller
- 1995
(Show Context)
Citation Context ...Paris and Vencovská (1997) offer the following resolution. They argue that the maximum entropy principle has been misapplied in this type of example: if 41 (Keynes, 1921) 42 (Keynes, 1921, §4.21) 43 (=-=Halpern and Koller, 1995-=-, §1) 17san agent refines the propositional variable c into r∨b∨g one should consider not L ′ but L ′′ = {c, r, b, g} and make the agent’s knowledge, namely c ↔ r ∨ b ∨ g, explicit. If we do that then... |

19 | Towards a Philosophy of Real Mathematics - Corfield - 2003 |

18 |
On the infinite
- Hilbert
- 1925
(Show Context)
Citation Context ...ericchi (2001). 54 Subjectivists usually slip in a few further constraints: e.g., known truths must be given probability 1, and degrees of belief should be updated by Bayesian conditionalisation. 55 (=-=Hilbert, 1925-=-) 25sinterest. Are probabilities uniquely determined, independently of background knowledge? If two agents disagree as to probabilities must at least one of them be wrong, even if they disagree as to ... |

17 |
Causality and maximum entropy updating
- Hunter
- 1989
(Show Context)
Citation Context ...ive conclusion. Keynes himself had stressed the importance of taking qualitative knowledge into account and the difficulties that ensue if qualitative information is ignored: 48 (Pearl, 1988, p. 468; =-=Hunter, 1989-=-) 49 (Hunter, 1989, p. 91) 21sBernoulli’s second axiom, that in reckoning a probability we must take everything into account, is easily forgotten in these cases of statistical probabilities. The stati... |

15 |
A World of Propensities. Thoemmes
- Popper
- 1990
(Show Context)
Citation Context ...y and that can be used to measure the chance. 23 Thus 18Note that some authors use ‘propensity’ to cover a physical chance interpretation as well as the propensity interpretation discussed above. 19 (=-=Popper, 1990-=-, p. 17) 20 (Miller, 1994, p. 186) 21Lewis (1980, p. 99). See §§10, 20. 22 (Fetzer, 1982, p. 195) 23 (Popper, 1990, p. 17) 9sif the set of conditions relevant to car AB01 CDE breaking down that hold o... |

13 |
Critical rationalism: A restatement and a defence
- Miller
- 1994
(Show Context)
Citation Context ... measure the chance. 23 Thus 18Note that some authors use ‘propensity’ to cover a physical chance interpretation as well as the propensity interpretation discussed above. 19 (Popper, 1990, p. 17) 20 (=-=Miller, 1994-=-, p. 186) 21Lewis (1980, p. 99). See §§10, 20. 22 (Fetzer, 1982, p. 195) 23 (Popper, 1990, p. 17) 9sif the set of conditions relevant to car AB01 CDE breaking down that hold on January 1st 2005 also h... |

13 | in Objective Bayesian Nets - Williamson |