Rationality and Objectivity in Science or Tom Kuhn meets Tom Bayes

Transcription

1 Rationality and Objectivity in Science or Tom Kuhn meets Tom Bayes Three (Ambitious) Objectives: 1. Partial rapprochement of Kuhnian ideas (with their subjective element) and logical empiricist approach to theory choice, via Bayes theorem. partial: only later Kuhn (Postscript + O, VJ & TC), not SSR key to rapprochement: analysis of plausibility arguments allows us to preserve context of discovery vs. context of justification 2. Defense of objective Bayesianism (= frequentist interpretation of the probabilities in Bayes Theorem). fall-back position: tempered personalism (personal probabilities constrained by some objective factors) 3. An account of rationality and objectivity in science. three grades of rationality (about current beliefs and belief revision): - static (consistency and coherence of current beliefs) - kinematic (Bayesian conditionalization as means of revision) - dynamic (objective probabilities + Bayesian algorithm for theory choice) objectivity = use objective probabilities in applying Bayes Theorem Idea: strive to eliminate subjective factors - priors: frequentist interpretation best guess of frequency of success - likelihoods: - given by logic, chance set-up, observed frequencies, or statistical significance tests - if not: plausible scenarios - expectedness/likelihood on catchall: - use likelihood ratios

2 1. Sections 1-3 i) Kuhn on theory choice: reacting mainly to inadequacies of H-D does not address SSR, ch. 12: explicitly anti-bayesian arguments based on incommensurability ii) Pro Kuhn: irreducibly subjective factors enter into theory choice, reflected in the prior probabilities and (perhaps) the likelihoods. Kuhn s five criteria (accuracy, consistency, scope, simplicity, fruitfulness) can be accommodated in the probabilities that make up Bayes Theorem, or in other theoretical virtues besides confirmation iii) Contra Kuhn: there is an algorithm of theory choice, Bayes Theorem a logic of science (given by Bayes Theorem), consistent with different probability assignments (or degrees of belief). too rigid? iv) Interpretation of probabilities: Salmon wants to interpret all the probabilities in Bayes Theorem as objective, i.e., based on frequencies. (p. 180) Alternative is personalism.

3 2. Prior probabilities (section 4) i) Identification of prior probability with plausibility plausibility = probability prior to any testing importance of plausibility arguments: limitations on investigation ii) Inadequacy of purely subjective priors orthodox personalist imposes only coherence: priors can be utterly irrational and idiosyncratic want to exclude irrelevant factors in estimating prior probability: the goal is to approximate frequencies iii) Tempered personalism: coherence + open-mindedness (avoid assigning 0 and 1) + assigning non-negligible prior to any hypothesis seriously advanced by serious scientists iv) Identification of prior probability with the (finite) frequency with which theories like T are true prior represents the best estimate of the chance of success of the hypothesis on the basis of all relevant experience (p. 186); or best estimates of the frequencies with which certain kinds of hypotheses succeed (187). assess this frequency using: - pragmatic criteria (serious scientist or crank?) - formal criteria (internal consistency; compatibility with accepted laws) - material criteria: analogy (success with similar hypotheses), simplicity (supplied by training and experience), symmetry Against a frequentist reading: how can we make sense of singlecase priors in terms of frequencies? What is the frequency that hypotheses like the Copernican hypothesis are true? fall-back position: tempered personalism no serious disagreement between this and the frequentist interpretation (?)

4 3. Expectedness and Likelihood (sections 5-7) i) Expectedness: two problems a) Likelihood problem (affects both objective (frequentist) and tempered personalist approach): how assign P(E / B)? - use of Bayes Theorem is not compatible with total naivety about T - generally have to expand denominator and use likelihoods b) Triviality problem (affects frequentist approach): P(E / B) = 1, since B includes experimental set-up, and then E is bound to occur. ii) Likelihoods a) Unproblematic cases: - P(E / T B) = 0 or 1 due to logical relationship - chance set-up - frequency data b) Problematic cases: e.g., absence of stellar parallax c) Catch-all hypothesis: T n in third version of Bayes Theorem stands for none of the above. How estimate P(E / T n B)? iii) Solution: use likelihood ratios P(T 1 / E B) = P(T 1 / B) P(E / T 1 B) P(T 2 / E B) P(T 2 / B) P(E / T 2 B) Give up on finding absolute value of posterior In this equation, the catch-all (or the expectedness) disappears We still have a comparative concept of confirmation: you should change your preference if the inverse ratio of likelihoods exceeds the ratio of priors

5 i) Auxiliary hypotheses Objection: 4. Plausible Scenarios (section 8) Problem: - even if we use likelihood ratios, we still need to deal with likelihoods on the hypotheses under consideration a) there is often no meaningful way to evaluate these (and preserve the spirit of either an objective or tempered personalist approach); or, b) the likelihoods are often unacceptably high or low for your pet theory Solution: - introduce scenarios or auxiliary hypotheses that either a) allow us to evaluate the probability, or b) shift the conditional probability value in a desired direction New likelihood ratio: Ex: P (observable parallax / Copernican hypothesis) 1 But P(obs. Parallax / Copernican & far away) 0 P(A 1 T 1 / E B) = P(A 1 T 1 / B) P(E / A 1 T 1 B) P(A 2 T 2 / E B) P(A 2 T 2 / B) P(E / A 2 T 2 B) The debate now shifts to take account of the combined priors, and the new likelihoods. 1) No help in cases of scientific revolution, since assessments of the priors will differ. 2) Broader worries about incommensurability: - different languages (no common set of priors ; no reason to assume Ptolemaic astronomers even have a prior probability for the Copernican hypothesis) - issues other than confirmation relevant to theory choice (e.g., perceived importance of explained phenomena) Has Salmon addressed Kuhn s worries (in earlier SSR)?

6 5. Kuhn s criteria (section 9) Kuhn s list: scope, accuracy, simplicity, consistency, fruitfulness. Kuhn s objection to algorithm for theory choice: subjective variation in 1) interpretation/application of the criteria; 2) relative weighting or importance of the criteria. Does Salmon address these points? - acknowledges that confirmation cannot be the only basis for theory choice (else we d stick to trivial theories). - informational, economic, confirmational virtues: but he offers no weighting for these, and says little about the first two 1. Scope: - Kuhn unclear (incommensurability incompatible with scope comparison) - Salmon offers no account 2. Accuracy: - informational/economic interpretation * alternative: confirmational interpretation (expectedness) 3, 4. Simplicity, consistency: - interpretation in terms of prior probabilities - latitude for subjective influence: but tempered personalism + convergence should ensure eventual agreement 5. Fruitfulness: - wide scope (translates into higher prior?) - able to explain unexpected phenomena (low P(E)) - gives rise to plausible scenarios Summary: a program that suggests some ways to mitigate Kuhn s arguments.

7 6. Rationality vs. Objectivity (section 10) 1) Three grades of rationality: a) Static: avoid logical contradictions and incoherent probabilities. (minimal connection to objectivity only the objectivity of logic and mathematics is needed) b) Kinematic: static, plus add the rule of Bayesian conditionalization for updating beliefs. This is just the orthodox personalist (Bayesian) position. (still a minimal connection to objectivity) c) Dynamic: kinematic, plus objective priors in Bayes theorem (fall-back: tempered personal priors). 2) Conclusions - tenability depends upon tenability of Salmon s objective probabilities - the fall-back position incorporates a blend of objective and subjective factors in formulating priors, likelihoods distinction between context of discovery and context of justification can be maintained Kuhnian emphasis on scientific judgement is correct, but we can locate such judgements within a rational context: a Bayesian approach to confirmation