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Bayesian Epistemology Kareem Khalifa Department of Philosophy Middlebury College The twin pillars of Bayesianism • Inductive reasoning can be represented in terms of probability theory: – Probability places coherence constraints on rational degrees of belief – Inductive reasoning proceeds through a probabilistic rule of inference called conditionalization • A rule of reasoning is justified if it passes the “pragmatic self-defeat” test (Dutch Book) What is probability? • Several interpretations • Logical interpretation – Probability: approximation of deductive entailment/ truth • Frequency interpretation – Probability: how often frequently something happens • Propensity interpretation – Probability: a physical system’s disposition to produce a certain outcome • Subjective interpretation = Bayesian – Probability: Degree of belief Coherence • Property of a synchronic set of beliefs – Synchronic set of beliefs = beliefs held at one time • Ex. A theory • Criterion of rationality – Coherent set of beliefs = rational – Incoherent set of beliefs = irrational • Probabilistic coherence: does not lead to bad wagering policies (‘pragmatic self-defeat’) in the long run Conditionalization • Bayes Theorem: P(h/e) = probability of h given e – Ex. P (There is fire / There is smoke) = 0.9 means it’s generally true that where there’s smoke, there’s fire. • Simple Rule of Conditionalization: Pf(h) = Pi(h/e) – Pi = prior probability = the degree of belief that one has in a claim prior to collecting evidence e – Pf = posterior probability = the degree of belief that one has in a claim after collecting evidence e Conditionalization: “3 Stages” • Throw yourself into a situation; assign prior probabilities • Gather evidence; assign a posterior probability of 1 to your evidence • Conditionalize to find posterior of hypothesis Conditionalization: “Stage 1” • Stage 1: You are thrown into a world with degrees of belief in various claims (priors), including: – Pi(h/e) = probability a hypothesis is true given a certain piece of evidence – Ex. For an early Australian explorer • Pi (Platypuses exist / Platypuses are observed) = 0.9 Conditionalization: “Stage 2” • Stage 2: Gather the evidence. For all evidence gathered, assign a posterior probability Pf(e) = 1 (i.e., you are now certain that e is true) – Ex. You observe a platypus. Pf(Platypuses are observed) = 1 Conditionalization: “Stage 3” • Ascertain how well this new evidence confirms your hypothesis using the rule of conditionalization. • Pf(h) = Pi(h/e) • Ex. Pf (Platypuses exist) = Pi (Platypuses exist / Platypuses are observed) = 0.9 Conditionalization and Modus Ponens • Modus Ponens: Deductive Rule of Inference: – If e, then h –e – Therefore h • Conditionalization: Inductive Rule of Inference – If e, then probably h, i.e., Pi(e/h) = p. – e, i.e., Pf(e) = 1. – Therefore probably h, i.e., Pf(h) = Pi(e/h) = p. Bayesian Confirmation • Pf(h) = Pi(h/e) = Pi(e/h) x Pi(h) / Pi(e) – Corollary of Simple Conditionalization • Admits of degrees of confirmation • Three stages still obtain: – Throw yourself into a situation; assign prior probabilities – Gather evidence; assign a posterior probability of 1 to your evidence – Conditionalize to find posterior of hypothesis Bayesian confirmation: Stage 1 • More priors to be assigned: • Pi(e/h) = probability of evidence e given the truth of the hypothesis h – Called the likelihood of h on e – The converse of the prior examined in Simple Conditionalization • Pi(e) = probability that evidence obtains • Pi(h) = probability that hypothesis is true Bayesian confirmation: Stages 2 & 3 • Stage 2 = Gather evidence, assign Pf(e) = 1 – Just as with Simple Conditionalization • Stage 3: Calculate Pf(h) = Pi(e/h)P(h)/Pi(e) – Like Simple Conditionalization, but more interesting – Less like Modus Ponens The Platypus Example • Stage 1: Assign priors – Pi(Platypus observed / Platypus exists) = .6 – Pi(Platypus observed) = .4 – Pi(Platypus exists) = .5 • Stage 2: Gather evidence – Pf(Platypus observed) = 1 • Stage 3: Calculate posterior for hypothesis – Pf(Platypus exists) = (.6)(.5)/(.4) = .75 Some Bayesian truisms • Evidence e confirms hypothesis h if and only if Pi(h/e) > Pi(h) • Evidence e disconfirms hypothesis h if and only if Pi(h/e) < Pi(h) • If h deductively entails e, then e confirms h and ~e disconfirms h (by reducing Pf(h) = 0) – Echoes HD and PF • Comparing hypotheses: – Pf(ha)/Pf(hb) = [Pi(e/ha) x Pi(ha)] / [Pi(e/hb) x Pi(hb) Intuitive consequences of Bayesianism • The higher the likelihood, the better confirmed the hypothesis • The higher the prior of the hypothesis, the better confirmed the hypothesis – Gives higher confirmation to hypotheses that are already strongly believed • The lower the prior of the evidence, the better confirmed the hypothesis – Privileges bold predictions Bayesian virtues • Demarcation • Induction – Underdetermination – Raven Paradox – Grue • Induction and Decision • Vagueness Bayesian Demarcation • Both laypeople and scientists can be treated as Bayesians • Differences between laypeople and scientists: – Evidence – Testing – How priors are assigned (possibly) Bayesians vs. Underdetermination • Underdetermination of theory by data: Many hypotheses can entail the same set of data (such hypotheses are empirically equivalent) • However, not all of them are equally probable given that data Bayesians vs. Raven Paradox • Bayesian solution: difference in likelihoods • Pi(non-black, non-raven / all ravens are black) would be low • Pi(black raven / all ravens are black) would be high • So non-black, non-ravens don’t have equal confirmatory power for the hypothesis that all ravens are black. Bayesian Holism • Scientific testing consists of a complex conjunction of theoretical (“core”) and auxiliary hypotheses about the workings of instruments, measurements, definitions of concepts, etc. • Theoretical and auxiliary hypotheses (even empirical statements) can be: – Probabilistically dependent upon each other – Of differing probabilities • Thus, an experiment may more strongly disconfirm one statement than another Bayesians vs. Grue • Let h1= All emeralds are green. • Let h2= All emeralds are grue, i.e., green now, but will turn blue on May 10, 3776 (my 2000th birthday!) • Why choose h1 over h2? • The higher the prior of the hypothesis, the better confirmed the hypothesis. – Our prior for h1 is higher than our gruesome h2 hypothesis. Bayesian Precision • Popper, Kuhn, and Thagard all suffered from vagueness in their core concepts (falsification, paradigm, and the theoretical virtues) • Bayesianism does not appear to have this problem