Understanding Bayesian Reasoning: Unpacking Statistical Fallacies and Inductive Logic

1. Stupid Bayesian Tricks Gregory Lopez, MA, PharmD SkeptiCamp 2009

2. Outline • Bayesiwhat? • Examining inductive arguments • Examining a formal and informal fallacy • An example of a conditional probabilistic fallacy

3. What’s Bayesian epistemology? • A (controversial) way to describe the relationship between evidence and hypotheses • Useful for induction and other instances of reasoning under uncertainty (probabilistically)

4. What does Bayesianism tell us about evidence? • Prediction principle: • e confirms h if p(e|h) > p(e|¬h) • Corollary: If h entails e, then e confirms h for anyone who does not already reject h or accept e • Thus, evidence that’s already known for sure does not confirm! • Discrimination principle: • If someone believes h more than h*, new evidence e cannot overthrow h unless p(e|h*) > p(e|h) • Note that the relative support for hypotheses depends on how well they predict the evidence under consideration • Surprise principle: • If a person is equally confident in e and e* conditional on h, then e confirms h more strongly for her than e* does (or disconfirms it less strongly) iff she is less confident of e than e* Joyce, JM. Bayesianism. In: Mele AR, Rawling P, Eds. The Oxford Handbook of Rationality. Oxford University Press, 2004

5. Practical applications of the principles to induction • Discrimination principle implies: • Similarity effect: • If you think that x is more similar to y than z and x it’s found that x has property P, then it’s more likely that y will be P than z • Typicality effect: • If x and y are members of a class but y is thought to be less typical, then getting data on x increases the probability of generalization more than getting data on y • Surprise principle implies: • Diversity effect: • When generalizing to a class, if property P holds amongst a diverse sample, it makes the generalization more probable than if the sample is less diverse Heit E. A Bayesian Analysis of Some Forms of Inductive Reasoning. In Rational Models of Cognition, M. Oaksford & N. Chater (Eds.), Oxford University Press, 1998.

6. Are fallacies always fallacious? • Formal fallacies: • Example: affirming the consequent • Informal fallacies: • Called informal because it has not been possible to give “a general or synoptic account of the traditional fallacy material in formal terms” • Example: argument from ignorance Hamblin, C. L. (1970). Fallacies. London: Methuen.

7. Affirming the consequent • If A then B, B; therefore, A • But doesn’t science work on this principle? • When working with this probabilistically, it can be seen as inference to the best explanation: • Only true if p(e|¬h) is low • Fails when there are multiple other plausible explanations or e is a strange event Korb, K. (2003). Bayesian informal logic and fallacy. Informal Logic, 24, 41–70.

8. Argument from ignorance • “There’s no evidence for x, so not x” • Increases as specificity increases and the prior decreases • “There’s no evidence that ghosts don’t exist, so they do!” vs. “There’s no evidence that vaccines cause autism, so they don’t!” Hahn, U., & Oaksford, M. (2007). The Rationality of Informal Argumentation: A Bayesian Approach to Reasoning Fallacies. Psychological Review, 114, 704-–32.

9. A conditional probability fallacy • Does order in the universe imply a god? • Assume that p(o|g) > p(o|¬g) • This isn’t what we want! We want the inverse! • However, p(g|o) > p(¬g|o) iff p(g) > p(¬g) • Therefore, order doesn’t imply a god unless we believe a god’s likely in the first place! Priest G. Logic: A Very Short Introduction. Oxford University Press. 2001

10. Discuss!