On Bayesian Measures 27 May 2005 V. Crup i Vincenzo Crupi Department of Cognitive and Education Sciences Laboratory of Cognitive Sciences University of Trento 27 May 2005 On Bayesian Measures of Evidential Support: Normative and Descriptive Considerations

On Bayesian Measures 27 May 2005 V. Crup i Core Bayesianism… (CB) confirmation is represented by a function X depending only on probabilistic information about evidence (premise) e and hypothesis (conclusion) h and identifying confirmation with an increase in the probability of h provided by the piece of information e, so that: X(e,h)X(e,h) > 0 iff p(h|e) > p(h) = 0 iff p(h|e) = p(h) < 0 iff p(h|e) < p(h)

On Bayesian Measures 27 May 2005 V. Crup i …and the plurality of Bayesian measures

On Bayesian Measures 27 May 2005 V. Crup i if p(h) = 0,5 D R L continues

On Bayesian Measures 27 May 2005 V. Crup i symmetries and asymmetries: an example Eells, E. & Fitelson, B. “Symmetries and Asymmetries in Evidential Support”, Philosophical Studies, 107 (2002), pp “evidence symmetry”: X(e,h) = –X(¬e,h) main thesis: the most adequate Bayesian measure(s) may be selected by testing competitors against intuitively compelling symmetries and asymmetries e.g.: “does a piece of evidence e support a hypothesis h equally well as e’s negation (¬e) undermines the same hypothesis h?” (p. 129) ?

On Bayesian Measures 27 May 2005 V. Crup i symmetries and asymmetries: a unified account a symmetry s is a function mapping an argument (e,h) onto a different argument by one or more of the following steps: negate e negate h invert premise and conclusion

On Bayesian Measures 27 May 2005 V. Crup i continues negate e negate h invert premise and conclusion  “evidence symmetry” (ES): ES(e,h) = (¬e,h)

On Bayesian Measures 27 May 2005 V. Crup i negate e negate h invert premise and conclusion  “hypothesis symmetry” (HS): HS(e,h) = (e,¬h) continues

On Bayesian Measures 27 May 2005 V. Crup i negate e negate h invert premise and conclusion  “inverse symmetry” (IS) such that: IS(e,h) = (h,e) continues

On Bayesian Measures 27 May 2005 V. Crup i negate e negate h invert premise and conclusion  “total symmetry” (TS): TS(e,h) = (¬e,¬h) continues

On Bayesian Measures 27 May 2005 V. Crup i negate e negate h invert premise and conclusion  “inverse evidence symmetry” (IES): IES(e,h) = (h,¬e) continues

On Bayesian Measures 27 May 2005 V. Crup i (e,h)(e,h) evidence (ES): (¬e,h) hypothesis (HS): (e,¬h) total (TS): (¬e,¬h) inverse (IS): (h,e) inverse evidence (IES): (h,¬e) inverse hypothesis (IHS): (¬h,e) inverse total (ITS): (¬h,¬e) convergent a symmetry s is: convergent iff (e,h) and s(e,h) have the same direction (both confirmations or both disconfirmations) divergent iff (e,h) and s(e,h) have opposite directions a convergent symmetry s holds iff X(e,h) = X[s(e,h)] a divergent symmetry s holds iff X(e,h) = –X[s(e,h)] continues

On Bayesian Measures 27 May 2005 V. Crup i Eells & Fitelson: an adequate measure of evidential support should violate “evidence symmetry” for at least some choice of e and h e.g.: X(Jack,face) >> –X(not-Jack,face) X(ace, face) << –X(not-ace,face) “the extremeness of logical implication [refutation] of (or conferring probability 1 [0] on) [the hypothesis] is not what is crucial to the examples for the purposes of evaluating […] (ES)” (p. 134) continues E & F suggest to extrapolate from extreme to non-extreme cases by simply assuming that the relevant premises describe reports of “very reliable, but fallible, assistants” (ibid.)

On Bayesian Measures 27 May 2005 V. Crup i V(e,h)V(e,h) = +1 iff e implies h = –1 iff e refutes h (i.e., implies not-h) = 0 otherwise principle of extrapolation (from the deductive to the inductive domain): (PE) any symmetry s holds for an adequate measure of evidential support iff it demonstrably holds for V continues

On Bayesian Measures 27 May 2005 V. Crup i (PE) implies that (IS) should not hold generally  counterexample: X(Jack,face) >> X(face,Jack) BUT (PE) also implies that (IS) should hold generally for pairs of disconfirmatory arguments X(e,h) – = X(h,e) – because e refutes h iff h refutes e (PE) implies that (ES) should not hold generally  counterexamples involving deductive arguments (PE) implies that (HS) should hold generally X(e,h) = –X(e,¬h) because e implies h iff e refutes ¬h continues

On Bayesian Measures 27 May 2005 V. Crup i the whole set of consequences of (PE): continues

On Bayesian Measures 27 May 2005 V. Crup i continues any Bayesian measure of evidential support fulfils the consequences of (PE) concerning (HS) and (IS) iff it fulfils all the consequences of (PE) concerning the other symmetries theorem:

On Bayesian Measures 27 May 2005 V. Crup i the whole set of consequences of PE: continues

On Bayesian Measures 27 May 2005 V. Crup i the need for yet another Bayesian measure of confirmation Eells & Fitelson (2002) suggest that measures D and L are to be preferred to other measures because both: fulfil (HS) violate (ES) violate (TS) violate (IS) [but E&F disregard the disconfirmation case…] however:

On Bayesian Measures 27 May 2005 V. Crup i continues  none of the currently available Bayesian measures of evidential support satisfies both (CB) and (PE) is it possible to define a measure of evidential support satisfying both (CB) and (PE) (and therefore the whole set of desirable symmetries and asymmetries)? existence proof:

On Bayesian Measures 27 May 2005 V. Crup i continues 0,5 p(h|e)p(h|e) 0,2 p(h|e)p(h|e) 0,8 p(h|e)p(h|e) measure Z

On Bayesian Measures 27 May 2005 V. Crup i continues Th 1. Z satisfies (HS) Proof: Z(e,h) + = 2/π arcsin{1 – [p(¬h|e)/p(¬h)]} = –2/π arcsin{[p(¬h|e)/p(¬h)] – 1} = – Z(e,¬h) – since h = ¬(¬h), this is equivalent to (HS) Th 2. Z satisfies (IS) for pairs of disconfirmatory arguments Proof: Z(e,h) – = 2/π arcsin{[p(h|e)/p(h)] – 1} = 2/π arcsin{[p(e|h)/p(e)] – 1} (by Bayes theorem) = Z(h,e) – Th 3. Z violates (IS) for some pair of confirmatory arguments Proof: suppose that: p(h & e) = 49/10 p(h & ¬e) = 41/10 p(¬h & e) = 1/100 p(¬h & ¬e) = 9/100 then: Z(e,h) + = 2/π arcsin(4/5) > 2/π arcsin(4/45) = Z(h,e) +

On Bayesian Measures 27 May 2005 V. Crup i empirical comparison of competing measures are the most normatively justified confirmation measures among the most psychologically descriptive? Tentori, K., Crupi, V., Bonini, N. & Osherson, D., “Comparison of Confirmation Measures”, 2005

On Bayesian Measures 27 May 2005 V. Crup i participants, materials and procedure: 26 students (Milan, Trento; mean age 24) 2 urns: (A) 30 black balls + 10 white balls (B) 15 black balls + 25 white balls random selection of one urn (outcome hidden) ten random extractions without replacement after each extraction:

On Bayesian Measures 27 May 2005 V. Crup i continues results : average correlations between judged evidential impact and confirmation measures (evidential impact computed from objective probabilities) each number is the average of 26 correlations (one per participant) for each correlation n = 10 p(A[B]|e) denotes p(A|e) or p(B|e) as appropriate. * = reliably greater than the average for p(A[B]|e) by paired t-test (p < 0,02)

On Bayesian Measures 27 May 2005 V. Crup i continues comparison of Z with other confirmation measures (confirmation computed from objective probabilities) each cell reports a paired t-test between the correlations obtained with the confirmation measures in the associated row and column for each t-test, n = 26 (corresponding to the 26 participants) the correlations each involve 10 observations the last row of each cell shows the number of participants (out of 26) for whom Z predicted better than the rival measure at the top of the column

On Bayesian Measures 27 May 2005 V. Crup i conclusive remarks limitations: a strictly probabilistic setting the results suggest a remarkable convergence between the normative and the descriptive dimension in the study of evidential support (compare with probability judgments!…) most symmetries and asymmetries were not involved in the experiment (e.g., IS); so IF it turned out that the consequences of (PE) are in fact reflected in na ï ve subjects’ judgments of evidential impact, THEN: – none of the available Bayesian measures could fully account for human inferential processes (not even in purely probabilistic settings) – Z (or similar measures) could have an even greater advantage against competing alternatives than the one detected in our study suggestions for further studies: – more direct comparisons – direct test of various symmetries and asymmetries – extension to non-probabilistic settings

On Bayesian Measures 27 May 2005 V. Crup i thank you