A graded Bayesian coherence notion Frederik Herzberg Center for - PowerPoint PPT Presentation

A graded Bayesian coherence notion Frederik Herzberg Center for Mathematical Economics (IMW), Bielefeld University Munich Center for Mathematical Philosophy, Ludwig Maximilian University of Munich W ORKSHOP ‘F ULL AND P ARTIAL B ELIEF ’ ( CO - LOCATED WITH THE 4 TH R ENÉ D ESCARTES L ECTURES ) Tilburg University 21 October 2014 imw Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 1 / 56

Introduction Introduction 1 2 Formalisation of (the core of) BonJour’s coherence notion Formal framework Formalising BonJour’s first desideratum Formalising BonJour’s second desideratum Formalising BonJour’s third and fourth desiderata Example: BonJour’s “ravens” challenge 3 Preparations for the formalisation of BonJour’s challenge Formalisation of the belief systems Calculation of the third and fourth component of the coherence measure Calculation of graded probabilistic consistency. The second component of the coherence measure Calculation of the first component of the coherence measure Summary Conclusion and discussion 4 imw Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 2 / 56

Introduction Introduction (1) Epistemic (doxastic) justification is a central and ancient topic of theoretical philosophy: ◮ Can one ever be justified in believing any proposition? ◮ If so, what are necessary and/or sufficient conditions ? Based on Aristotle’s “regress argument”, it has been argued that there are exactly three non-skeptical views about the structure of epistemic justification ( structure of reasons ). According to this position, non-skeptical positions on epistemic justification either assert ◮ that reasons form a finite chain with some foundational proposition at the base ( foundationalism ) — illustration: foundation of a house (D ESCARTES ); or ◮ that reasons mutually support each other ( coherentism ) — illustration: planks of a boat (N EURATH ); or ◮ that reasons form an infinite regress ( infinitism ) — illustration: open-ended loop. imw Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 3 / 56

Introduction Introduction (2) The philosophical discussion on epistemic justification has seen interesting turns during the past two decades: ◮ infinitist accounts of epistemic justification were revived (e.g. K LEIN 1998ff; formally: P EIJNENBURG 2007); ◮ a major proponent of coherentism abandoned this position (B ON J OUR 1999, 2010) for Cartesianism; ◮ impossibility theorems for coherence measures suggest that coherentism defies formalisation (pioneers: K LEIN –W ARFIELD 1994), formally reiterating an earlier criticism by E WING (1934). We give a formal defense of coherentism that takes traditional epistemology seriously: ◮ We propose a class of coherence measures that meet the thrust of BonJour’s desiderata . ◮ The domain of these coherence measures will be systems of degrees of belief . (Our own position is a graded version of sufficiency coherentism; it accommodates foundationalist and infinitist intuitions, but rejects imw the Principle of Inferential Justification. H. 2014b) Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 4 / 56

Introduction The (im)possibility of formal coherence concepts (1) There is a rich body of literature, originally closely related to the discussion on epistemological coherentism, on the (im)possibility of a formal graded coherence notion (K LEIN –W ARFIELD 1994, 1996; S HOGENJI 1999; A KIBA 2000; F ITELSON 2003; B OVENS –H ARTMANN 2003a,b, 2005, 2006; O LSSON 2002, 2005; D IETRICH –M ORETTI 2005; M EIJS –D OUVEN 2007; S CHUPBACH 2008; S IEBEL –W OLFF 2008). The overall finding of this literature is that it appears to be very difficult to come up with a convincing (especially one-dimensional) graded coherence notion . imw Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 5 / 56

Introduction The (im)possibility of formal coherence concepts (2) However, the recent formal literature on coherence measures ◮ does not interact closely with the traditional literature on coherentism (e.g. B ON J OUR 1985, L EHRER 2000) and ◮ in general models belief systems as sets of propositions endowed with a unique probability measure — a very strong assumption (psychologically and decision-theoretically less than compelling). We suggest a new formal framework which models belief systems as sets of conditional probability assignments , compatible with several (even infinitely many) probability measures; they induce a Bayesian network on the propositions. Within that framework, we propose a formalisation of the thrust of BonJour ’s (1985) (multi-dimensional) coherence concept : ◮ inferential connections and fragmentation are measured through graph-theoretic concepts on the induced Bayesian network; ◮ probabilistic consistency is measured via the size of the set of probability measures compatible with the belief system. imw (H. 2014c) Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 6 / 56

Introduction Criteria for a formal graded coherence notion (1) B ON J OUR ’s coherence concept provides desiderata for a formal coherence notion: [(I)] A system of beliefs is coherent only if it is logically consistent. [(II)] A system of beliefs is coherent in proportion to its degree of probabilistic consistency. [. . . ] [(III)] The coherence of a system of beliefs is increased by the presence of inferential connections between its component beliefs and increased in proportion to the number and strength of such connections. [(IV)] The coherence of a system of beliefs is diminished to the extent to which it is divided into subsystems of beliefs which are relatively unconnected to each other by inferential connections. imw (B ON J OUR 1985, Section 5.3, pp. 95, 98, 99) Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 7 / 56

Introduction Criteria for a formal graded coherence notion (2) There is, in addition, also a fifth desideratum , which however does not admit a natural formalisation. [(V)] The coherence of a system of beliefs is decreased in proportion to the presence of unexplained anomalies in the believed content of the system. (B ON J OUR 1985, Section 5.3, pp. 95, 98, 99) We propose a class of formal coherence concepts which satisfy the first four of B ON J OUR ’s desiderata — and ultimately might be restricted to satisfy a formalisation of the fifth desideratum, too. There are several classes of epistemic anomalies. A local anomaly might be a non-foundational belief with high degree of centrality . A global anomaly might be a belief whose omission from the belief system imw would result in a substantial reduction in complexity . Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 8 / 56

Introduction Multiplicity of subjective probability measures (priors) Our formal framework allows for belief systems that are compatible with multiple probability measures . This reflects the consensus of contemporary decision theory (and also psychology, cf. e.g. M INSKY 1986 or O RNSTEIN 1986): ◮ Building on work by E LLSBERG (1961) and G ILBOA –S CHMEIDLER (1989), decision making under multiple priors or probabilistic ambiguity ( uncertainty in the sense of K NIGHT 1921) is studied. ◮ There is a body of literature on probabilistic opinion pooling (e.g. M C C ONWAY 1981 and C OOKE 1991). ◮ The problem of aggregating probability measures can also be studied within a comprehensive, theory of aggregating propositional attitudes (D IETRICH –L IST 2011). Most recently, these results have been extended: ◮ A unified methodology for the theory of propositional-attitude aggregation has been proposed, via universal algebra (H. 2014d). ◮ The theory of probabilistic opinion pooling has been extended to infinite profiles of priors — a set-theoretically delicate problem imw which can be solved using ultrafilters (H. 2014a). Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 9 / 56

Formalisation of (the core of) BonJour’s coherence notion Introduction 1 2 Formalisation of (the core of) BonJour’s coherence notion Formal framework Formalising BonJour’s first desideratum Formalising BonJour’s second desideratum Formalising BonJour’s third and fourth desiderata Example: BonJour’s “ravens” challenge 3 Preparations for the formalisation of BonJour’s challenge Formalisation of the belief systems Calculation of the third and fourth component of the coherence measure Calculation of graded probabilistic consistency. The second component of the coherence measure Calculation of the first component of the coherence measure Summary Conclusion and discussion 4 imw Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 10 / 56