Stupid Bayesian Tricks Gregory Lopez, MA, PharmD SkeptiCamp 2009

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

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

What does Bayesianism tell us about evidence? Prediction principle: Prediction principle: e confirms h if p(e|h) > p(e|¬h) 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 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! Thus, evidence that’s already known for sure does not confirm! Discrimination principle: Discrimination principle: If someone believes h more than h*, new evidence e cannot overthrow h unless p(e|h*) > p(e|h) 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 Note that the relative support for hypotheses depends on how well they predict the evidence under consideration Surprise principle: 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* 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

Practical applications of the principles to induction Discrimination principle implies: Discrimination principle implies: Similarity effect: 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 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: 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 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: Surprise principle implies: Diversity effect: 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 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.

Are fallacies always fallacious? Formal fallacies: Formal fallacies: Example: affirming the consequent Example: affirming the consequent Informal fallacies: 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” 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 Example: argument from ignorance Hamblin, C. L. (1970). Fallacies. London: Methuen.

Affirming the consequent If A then B, B; therefore, A If A then B, B; therefore, A But doesn’t science work on this principle? But doesn’t science work on this principle? When working with this probabilistically, it can be seen as inference to the best explanation: When working with this probabilistically, it can be seen as inference to the best explanation: Only true if p(e|¬h) is low Only true if p(e|¬h) is low Fails when there are multiple other plausible explanations or e is a strange event 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.

Argument from ignorance “ There’s no evidence for x, so not x” “ There’s no evidence for x, so not x” Increases as specificity increases and the prior decreases 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!” “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.

A conditional probability fallacy Does order in the universe imply a god? Does order in the universe imply a god? Assume that p(o|g) > p(o|¬g) Assume that p(o|g) > p(o|¬g) This isn’t what we want! We want the inverse! This isn’t what we want! We want the inverse! However, p(g|o) > p(¬g|o) iff p(g) > p(¬g) 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! 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

Discuss!