Bayesian ModelingQ550: Models in Cognitive ScienceModeling Probabilistic Inference• Probability was originally developed as a means forstudying games of chance and making optimumgambling decisions• Scientific formalizations of probability are a fairly recentdevelopment• Jeffrey: Prior to the 17th C, probable meant approvable;a probable action or opinion was one that sensiblepeople would hold (given the circumstances)• More recently, we have used probability as a formalaccount of human decision making (e.g., Gigerenzer etal., 1989)Bounded Rationality• The assumption is that humans use bouned-rationaldecision strategies We attempt to optimize our decisions underconstraints imposed by human informationprocessing limitations Frequentist probability (Neyman, Venn, Fisher, etc.)• For this, we make good use of Bayes Theorem• Assume that human induction/learning/decision-makingis based on a Bayes engine: “Belief is goverened by thelaws of probability”Bayes Theorem• Bayes Theorem is a way to update beliefs in light of newevidence (a posteriori)• Humans are rational inference machines…we are testinghypotheses based on evidence from the environment• Assume you are trying to infer the process that wasresponsible for generating observable data (D). H is thehypothesis about the true generating process.• Assume you are trying to infer the process that wasresponsible for generating observable data (D). H is thehypothesis about the true generating process.By Bayes Theorem:! P(H | D) =P(D | H)P(H)P(D)• P(H|D) = posterior (Hyp given Data)• P(H) = prior• P(H|D) = (Data given Hyp) conditional or likelihood• P(D) = marginal or normalizing constant (ensures prob isnormalized to sum one)The posterior is proportional to the product of the priorprobability and the likelihoodAn example (from Griffiths & Yuille, 2006):• A box contains two coins: One that produces heads 50%of the time and one that produces heads 90% of the time• You pick a coin and flip it 10 times; how do you modifyyour hypothesis based on the data?Let θ = P(heads)We need to identify the hypothesis space H, the priordistribution, P(H), and the likelihood P(D|H)H0 is θ = 0.5 and H1 is θ = 0.9. P(H0) = P(H1) = 0.5The probability of a particular sequence is binomial:! P(D |") ="NH1#"NT( )With only two hypotheses, it is convenient to just work withthe posterior odds (ratio of evidence for H0 over H1):This is exactly as we did for model selection with the BICSequence of data: HHHHHHHHHH = 357:1 in favor of H1Sequence of data: HHTHTHTTHT = 165:1 in favor of H0! P(H1| D)P(H0| D)=P(D | H1)P(D | H0)P(H1)P(H0)Back to the Goat Game• Recall our earlier “Goat Game” (aka, Monte Hall 3-doorproblem).Hypothesis that car is behind a door: H1, H2, H3 P(H1) = P(H2) = P(H3) = 1/3• Assume that we pick door #1• Let D = “host opens door #2” Without prior knowledge,we would assign this event p = .5• We don’t even need to see what’s behind door #2. If thecar was there, he had to pick door 3. If the car wasbehind door #3, he had to pick door #2. Thus, given thisevidence (D), we know:! P(D | H2) = 0P(D | H3) = 1P(D | H1) = 1/2Thus, you should always switch from the original door you picked:! P(D | H2) = 0P(D | H3) = 1P(D | H1) = 1/2! P(H1| D) =P(D | H1)P(H1)P(D)=12"1312=13P(H2| D) =P(D | H2)P(H2)P(D)=0 "1312= 0P(H3| D) =P(D | H3)P(H3)P(D)=1"1312=23Bounded Rationality• Of course, most humans you’ll meet think the stick andswitch strategies are equally optimal Thus, humans are not optimal Bayesian decisionmakers, but they still may be within the restrictions ofbounded rationality (cf. Tversky & Kahneman’sdecision heuristics)• Choosing between infinitely many hypotheses (e.g.,language learning)…we get a probability density over arandom variable, and can use the maximum or meanposterior from the distribution (see Griffiths & Yuilletutorial on probabilistic inference)• Markov models and Bayes nets (graphical distributions)Deal or No Deal?• Task: Create a Bayesian model that optimizeshypotheses about the amount in cases based on thebehavior of the banker over trials…can you predictoptimum choices?• Distribution of payouts• Synthetic RPS competition• Who wants to be a millionaire w/ semantic modelsExamples of Bayesian Models:Tenenbaum, J. B. (1999). Bayesian modeling of humanconcept learning. NIPS• Humans are able to learn efficiently from a small numberof positive examples, and can successfully generalizewithout experiencing negative examples (e.g., wordmeanings, perceptual categories)• Poverty of the stimulus problem• By contrast, ML techniques require both positive andnegative examples to generalize at all (and manyexamples of each to generalize successfully)Examples of Bayesian Models:• Bayesian inference provides an excellent model of howhumans induce and generalize structure in theirenvironments They succeed where many other models fail atlearning flexible tasks as humans do.Other Examples:• Perfors, Tenenbaum, & Regier (2006)• Anything by Dan Navarro• The Topics model (Grif, Steyv, & Ten, in press, PR)• Chater & Manning (2006)• Kruschke (2006)• For a good overview, see Tenenbaum, Griffiths, & Kemp(2006) TICSAn application of modeling: Mike Mozer’s crazy