## Inductive influence

Venue: | British Journal for the Philosophy of Science |

Citations: | 9 - 7 self |

### BibTeX

@ARTICLE{Williamson_inductiveinfluence,

author = {Jon Williamson},

title = {Inductive influence},

journal = {British Journal for the Philosophy of Science},

year = {},

volume = {58}

}

### OpenURL

### Abstract

Objective Bayesianism has been criticised for not allowing learning from experience: it is claimed that an agent must give degree of belief 1 to the next raven being black, however many other black ravens have 2 been observed. I argue that this objection can be overcome by appealing to objective Bayesian nets, a formalism for representing objective Bayesian degrees of belief. Under this account, previous observations exert an inductive influence on the next observation. I show how this approach can be used to capture the Johnson-Carnap continuum of inductive methods, as well as the Nix-Paris continuum, and show how inductive influence can

### Citations

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(Show Context)
Citation Context ...8) 7 Note that Paris (1994, pp. 198–199) offers a similar diagnosis but a different resolution. See also Paris and Vencovská (2003). 8 See also Williamson (2002) and Williamson (2005a, §§5.6–5.7). 9 (=-=Pearl, 1988-=-; Neapolitan, 1990) 4s✓✏ B1 ✒✑ ✓✏ B2 ✒✑ ... Figure 1: No background knowledge. ✓✏ Bk ✒✑ Given a set of quantitative constraints involving the variables defined from an agent’s language, an objective B... |

2193 |
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(Show Context)
Citation Context ...e if the underlying data-generating process is an independent process, but a bad move if it is dependent. The objective Bayesian, on the other hand, does not need to 17 (Gillies, 2000, pp. 77–83) 18 (=-=Feller, 1950-=-, pp. 67–95; Popper, 1957; Gillies, 2000, pp. 77–83) 9sfully determine a prior at the outset, because she she updates as follows: on learning b ε1 1 bε2 2 · · · bεn n her new probability p ′ (b 1 n+1)... |

586 | Probability Theory: The Logic of Science
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(Show Context)
Citation Context ...e see that the Nix-Paris continuum of inductive methods emerges as another special case—though arguably a less central special case. The question now arises as to which point in 1 (Rosenkrantz, 1977; =-=Jaynes, 2003-=-) 2 I will only be considering finite probability spaces in this paper. The extension of objective Bayesianism to the infinite case is steeped in controversy and arguably proceeds at the expense of un... |

293 | Scientific Reasoning: The Bayesian Approach - Howson, Urbach - 1993 |

205 |
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(Show Context)
Citation Context ...t Paris (1994, pp. 198–199) offers a similar diagnosis but a different resolution. See also Paris and Vencovská (2003). 8 See also Williamson (2002) and Williamson (2005a, §§5.6–5.7). 9 (Pearl, 1988; =-=Neapolitan, 1990-=-) 4s✓✏ B1 ✒✑ ✓✏ B2 ✒✑ ... Figure 1: No background knowledge. ✓✏ Bk ✒✑ Given a set of quantitative constraints involving the variables defined from an agent’s language, an objective Bayesian net can be... |

164 |
The Estimation of Probabilities. An Essay on Modern Bayesian Methods
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(Show Context)
Citation Context ... inductiveinfluence framework is of key importance for the following reason. In their derivation of their continuum, Johnson and Carnap make a crucial assumption: 15 (Johnson, 1932; Carnap, 1952) 16 (=-=Good, 1965-=-, p. 18) 8s1 0.75 0.5 0.25 0 pε pε 50 40 30 λ 20 10 0 0 10 8 6 4 r10 2 Figure 4: The Johnson-Carnap continuum of inductive methods. a kind of exchangeability assumption. This is the assumption that th... |

102 |
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(Show Context)
Citation Context ... generality of the inductiveinfluence framework is of key importance for the following reason. In their derivation of their continuum, Johnson and Carnap make a crucial assumption: 15 (Johnson, 1932; =-=Carnap, 1952-=-) 16 (Good, 1965, p. 18) 8s1 0.75 0.5 0.25 0 pε pε 50 40 30 λ 20 10 0 0 10 8 6 4 r10 2 Figure 4: The Johnson-Carnap continuum of inductive methods. a kind of exchangeability assumption. This is the as... |

97 | Bayes or bust - Earman - 1996 |

76 |
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(Show Context)
Citation Context ...-Carnap continuum. This is fine if the underlying data-generating process is an independent process, but a bad move if it is dependent. The objective Bayesian, on the other hand, does not need to 17 (=-=Gillies, 2000-=-, pp. 77–83) 18 (Feller, 1950, pp. 67–95; Popper, 1957; Gillies, 2000, pp. 77–83) 9sfully determine a prior at the outset, because she she updates as follows: on learning b ε1 1 bε2 2 · · · bεn n her ... |

64 | The Uncertain Reasoner’s Companion - Paris - 1994 |

55 | Bayesian nets and causality: Philosophical and computational foundations - Williamson - 2005 |

32 |
Probability: The Deductive and Inductive Problems
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(Show Context)
Citation Context ...oach. The extra generality of the inductiveinfluence framework is of key importance for the following reason. In their derivation of their continuum, Johnson and Carnap make a crucial assumption: 15 (=-=Johnson, 1932-=-; Carnap, 1952) 16 (Good, 1965, p. 18) 8s1 0.75 0.5 0.25 0 pε pε 50 40 30 λ 20 10 0 0 10 8 6 4 r10 2 Figure 4: The Johnson-Carnap continuum of inductive methods. a kind of exchangeability assumption. ... |

24 | Motivating Objective Bayesianism: from Empirical Constraints to Objective Probabilities - Williamson |

22 |
Method and Decision: Towards a Bayesian Philosophy of Science
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- 1977
(Show Context)
Citation Context ... case (§6). In §7 we see that the Nix-Paris continuum of inductive methods emerges as another special case—though arguably a less central special case. The question now arises as to which point in 1 (=-=Rosenkrantz, 1977-=-; Jaynes, 2003) 2 I will only be considering finite probability spaces in this paper. The extension of objective Bayesianism to the infinite case is steeped in controversy and arguably proceeds at the... |

13 | Objective Bayesian Nets - Williamson |

12 |
A Continuum of inductive methods arising from a generalized principle of instantial relevance
- Nix, Paris
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(Show Context)
Citation Context ...raints (where θ, ϕ, ψ are quantifier-free sentences of a monadic first order predicate language containing infinitely many predicates): 22 19 See Good (1965, p. 17) for example, or Zabell (1982). 20 (=-=Nix and Paris, 2007-=-, Theorem 14) 21 (Nix, 2005, Proposition 4.2) 22 (Nix and Paris, 2007, Theorem 24) 10s1 0.75 0.5 0.25 0 pε pε 2 3 β 4 5 6 0 10 8 6 4 r10 2 Figure 5: The Nix-Paris continuum of inductive methods. Regul... |

10 |
Some observations on induction in predicate probabilistic reasoning
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(Show Context)
Citation Context ...ge in different directions. The game of twenty questions shows us that for a language to be optimal with respect to classification, each predicate should bisect the population of individuals: the 25 (=-=Wilmers et al., 2002-=-, Theorem 3) 12sproportion of individuals that instantiate a conjunction of j property terms should be 1/2j for j ≥ 1. In a language that is optimal for classification, 20 property terms suffice to un... |

9 |
Probabilistic induction in the predicate calculus
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(Show Context)
Citation Context ...free sentences of a monadic first order predicate language containing infinitely many predicates): 22 19 See Good (1965, p. 17) for example, or Zabell (1982). 20 (Nix and Paris, 2007, Theorem 14) 21 (=-=Nix, 2005-=-, Proposition 4.2) 22 (Nix and Paris, 2007, Theorem 24) 10s1 0.75 0.5 0.25 0 pε pε 2 3 β 4 5 6 0 10 8 6 4 r10 2 Figure 5: The Nix-Paris continuum of inductive methods. Regularity: p(θ) = 0 iff |= ¬θ. ... |

9 | Philosophies of Probability: Objective Bayesianism and its Challenges - Williamson - 2009 |

8 | The emergence of reasons conjecture - Paris, Vencovská - 2003 |

6 | 2007. Objective Bayesian nets for systems modelling and prognosis in breast cancer - Nagl, Williams, et al. |

6 | Maximising entropy efficiently - Williamson - 2002 |

5 | A critique of Jaynes’ maximum entropy principle - Dias, Shimony - 1981 |

5 | Objective Bayesianism with predicate languages, Synthese 163 (3 - Williamson - 2008 |

1 |
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- Popper
- 1957
(Show Context)
Citation Context ...generating process is an independent process, but a bad move if it is dependent. The objective Bayesian, on the other hand, does not need to 17 (Gillies, 2000, pp. 77–83) 18 (Feller, 1950, pp. 67–95; =-=Popper, 1957-=-; Gillies, 2000, pp. 77–83) 9sfully determine a prior at the outset, because she she updates as follows: on learning b ε1 1 bε2 2 · · · bεn n her new probability p ′ (b 1 n+1) is determined by the max... |