Gregory Wheeler Philosophical Foundations of Imprecise Probability ISIPTA 2015 Tutorial PHILOSOPHY MUNICH CENTER FOR MATHEMATICAL PHILOSOPH MATHEMATICAL PHILOSOPHY MUNICH CENTER FOR MATHEMATI MUNICH CENTER FOR MATHEMATICAL PHILOSOPHY MUNICH CE LUDWIG- L M U MAXIMILIANS- CAL PHILOSOPHY MUNICH CENTER FOR MATHEMATICAL PHILOS UNIVERSITÄT MÜNCHEN
william james The history of philosophy is to a great extent that of a certain clash of human temperaments. … [Temperament] loads the evidence … one way or the other, making for a more sentimental or a more hard-hearted view of the universe, just as this fact or that principle would (James 1907, 8–9). 1
william james The history of philosophy is to a great extent that of a certain clash of human temperaments. … [Temperament] loads the evidence … one way or the other, making for a more sentimental or a more hard-hearted view of the universe, just as this fact or that principle would (James 1907, 8–9). 1
reduction of numbers to sets Zermelo-Fraenkel Von Neumann 2 0 = {} 0 = {} 1 = { 0 } = {{}} 1 = { 0 } = {{}} 2 = { 1 } = {{{}}} 2 = { 0 , 1 } = {{} , {{}}} 3 = { 2 } = {{{{}}}} 3 = { 0 , 1 , 2 } = {{} , {{}} , {{} , {{}}}}
interpretations of probability Subjective Interpretation Ramsey (1926), de Finetti (1937), Savage (1954), Anscombe-Aumann (1963) Jeffreys (1939), Fisher (1936) Logical Interpretation Carnap (1945, 1952), Paris & Vencovská (2015), Kyburg (1961, 2001) Frequency / Propensity Interpretation 1 See (Glymour and Eberhardt 2012). 3 Reichenbach 1 and Popper (1959)
interpretating probability What is probability? Any response should answer at least three questions (Salmon 1967): 1. Why should probability have particular mathematical properties ? 2. How do are probabilities determined or measured ? 3. Why and when is probability useful ? 4
2. How are logical probabilities measured? logical probability 3. Why is logical probability useful ? Measures the strength of evidential support. Carnap Kyburg ? From statistical data 1. Why does logical probability satisfy the axioms ? Carnap Kyburg Analytic ? 5
logical probability 3. Why is logical probability useful ? Measures the strength of evidential support. Carnap Kyburg ? From statistical data 1. Why does logical probability satisfy the axioms ? Carnap Kyburg Analytic ? 5 2. How are logical probabilities measured?
logical probability 3. Why is logical probability useful ? Measures the strength of evidential support. Carnap Kyburg ? From statistical data 1. Why does logical probability satisfy the axioms ? Carnap Kyburg Analytic ? 5 2. How are logical probabilities measured?
subjective probability 3. Why is subjective probability useful ? Measures the strength of partial belief. 2. How is subjective probability measured ? Betting behavior Accurate forecasting Preferences among compound lotteries Preferences among acts 6 Allows us to calculate expected utility calculations.
subjective probability 3. Why is subjective probability useful ? Measures the strength of partial belief. Betting behavior Accurate forecasting Preferences among compound lotteries Preferences among acts 6 Allows us to calculate expected utility calculations. 2. How is subjective probability measured ?
subjective probability Preferences among lotteries Satisfy Probability Axioms. Rationality criteria then for the corresponding rationality criteria : Theorem: If probability is elicited via the measurement procedure , Savage axioms Preference among acts VNM & Anscombe-Aumann axioms Qualitative probability axioms 1. Why does subjective probability satisfy the axioms ? Qualitative verbal comparisons Minimizing squared-error loss Accurate forecasting Avoiding sure loss Betting behavior Rationality Criteria Measurement Procedure 7
subjective probability 1. Why does subjective probability satisfy the axioms ? then for the corresponding rationality criteria : Theorem: If probability is elicited via the measurement procedure , Savage axioms Preference among acts VNM & Anscombe-Aumann axioms Preferences among lotteries Qualitative probability axioms Qualitative verbal comparisons Minimizing squared-error loss Accurate forecasting Avoiding sure loss Betting behavior Rationality Criteria Measurement Procedure 7 Rationality criteria ⇔ Satisfy Probability Axioms.
epistemic decision theory Measurement Procedure Rationality Criteria Accurate forecasting Minimizing squared-error loss Alethic Property Rationality Criteria Gradational (in)accuracy ‘Distance’ from the truth 2 See (Joyce 1998; Joyce 2009; Leitgeb and Pettigrew 2010; Pettigrew 2013). 8 · Traditional dF-style use of (strictly) proper scoring rules: · Purely epistemic interpretation of (strictly) proper scoring rules: 2
back to james Two philosophical temperaments: Tender-minded: cling to the belief that facts should be related to values and that values seen as predominant. Tough-minded: want facts to be dissociated from values and left to themselves. 9
ip and scoring rules Joyce’s Commitments (1998, 2009) 10 · Credal commitments (belief) modeled by IP & · Purely epistemic interpretation of (strictly) proper scoring rules:
impossibility theorem Theorem (Seidenfeld et al. (2012) 3 ) Admissibility, Imprecision, Continuity, Quantifiability, Extensionality, and Strict Immodesty are jointly inconsistent. 3 A mild mathematical generalization is in Mayo-Wilson and Wheeler, forthcoming. 11
scoring rule For the purposes of this talk, 12 a scoring rule I ( b , ω ) denotes the ‘inaccuracy’ of the belief b about a proposition ϕ when the truth-value of ϕ is ω ∈ { 0 , 1 } .
plan Claim: there is no strictly proper IP scoring rule. Plan: give 6 necessary postulates that cannot all be satisfied. 13
postulates states, and suppose that d is at least as accurate as c whatever the truth. If your belief state is b and the set of rational belief contains d . 14 Admissibility Let b , c , and d be three (not necessarily distinct) belief states R b from your perspective contains c , then it also
Imprecision: A belief state is a set of real numbers between 0 and 1. Quantifiability: Degrees of inaccuracy are represented by non-negative real numbers. representing how inaccurate belief b is. Moreover, this degree depends only upon b and the Strict Immodesty: If your belief state is b , then the set of rational 15 Extensionality: For every truth-value ω and every belief state b , there is a single degree of inaccuracy I ( b , ω ) truth-value ω of the proposition ϕ of interest. belief states R b from your perspective is { b } .
pareto constraint Problem : It is unclear how to represent the distance between arbitrary sets of numbers between 0 and 1. How “close” are the beliefs that 7 ? 16 (i) a flipped coin lands heads in the interval [ 1 4 ] , and 4 , 3 (ii) a flipped coin lands heads in the interval [ 1 4 ] other 4 , 3 than 4
pareto constraint 0 c b a Suppose that belief states a , b , c are such that 1 17 Example: to be at least as great as the distance between the Constraint P The distance between the belief states a and c ought a − ≤ b − ≤ c − or a − ≥ b − ≥ c − , and a + ≤ b + ≤ c + or a + ≥ b + ≥ c + . belief states a and b . b − b + a − c − a + c +
Continuity Sufficiently similar belief states are similarly continuous with respect to the parameter b , where the metric on beliefs satisfies Constraint P . 18 inaccurate. More precisely, for all ω , the function I ( b , ω ) restricted to the set of interval beliefs b is
impossibility theorem Theorem (Seidenfeld et al. (2012) 4 ) Admissibility, Imprecision, Continuity, Quantifiability, Extensionality, and Strict Immodesty are jointly inconsistent. 4 A mild mathematical generalization is in Mayo-Wilson and Wheeler, forthcoming. 19
impossibility theorem: scope One to many propositions: Although formulated for a single proposition, our result extends to finitely many propositions with additional mathematical machinery to ensure the topological invariance of dimension. 5 Other Uncertainty Models: The theorem applies to Dempster-Shafer Belief functions and Ranking functions. 5 Thanks here to Catrin Campbell-Moore. 20
impossibility theorem: scope One to many propositions: Although formulated for a single proposition, our result extends to finitely many propositions with additional mathematical machinery to ensure the topological invariance of dimension. 5 Other Uncertainty Models: The theorem applies to Dempster-Shafer Belief functions and Ranking functions. 5 Thanks here to Catrin Campbell-Moore. 20
the options Central to accuracy-first epistemology Imprecision Central to IP Continuity Quantifiability Strict Immodesty Remaining options 21 · Admissibility · Extensionality
the options Central to accuracy-first epistemology Central to IP Continuity Quantifiability Strict Immodesty Remaining options 21 · Admissibility · Extensionality · Imprecision
the options Central to accuracy-first epistemology Central to IP Remaining options 21 · Admissibility · Extensionality · Imprecision · Continuity · Quantifiability · Strict Immodesty
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