Probabilistic Reasoning PDF

 Memory: Storage and Retrieval Storage and Retrieval: Random Access Memory Random Access Memory Storage and Retrieval: Random Access Memory Address Content 1 0 1 1 0 0 1 1 1 1 0 1 0 1...

 Memory: Storage and Retrieval Storage and Retrieval: Random Access Memory Random Access Memory Storage and Retrieval: Random Access Memory Address Content 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 Input: (1, 1) Output: (1, 1, 0, 0) Storage and Retrieval: Random Access Memory Address 1 0 1 1 0 1 1 1 The addresses can be systematic Storage and Retrieval: Random Access Memory Storage and Retrieval: Content-addressable memory Address-Content 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 Input: (1, 1, X, 1, 0, 1) Output: (1, 1, 0, 1, 0, 1) Input: (1, 1, 0, 1, 0, 1) Output: (1, 1, 0, 1, 0, 1) Storage and Retrieval: Content-addressable memory “An exquisite pleasure had invaded my senses, but individual, detached, with no suggestion of its origin.” “And suddenly the memory returns. The taste was that of the little crumb of madeleine which on Sunday mornings at Combray…, when I went to say good day to her in her bedroom, my aunt Leonie used to give me, dipping it first in her own cup of real or of lime-flower tea.” Marcel Proust Storage and Retrieval Godden and Baddeley 1975 Storage State-dependent retrieval Condition Mean number of words recalled Study Test Free recall Cued recall sober sober 11.5 24.0 drug sober 6.7 22.6 sober drug 9.9 23.7 drug drug 10.5 22.3 Eich et al. 1975 Probabilistic Reasoning Sources of Uncertainty Sources of Uncertainty Uncertainty about Perception Would you cross the street? Sources of Uncertainty Uncertainty about Perception Is there an animal on the tree? Sources of Uncertainty Uncertainty about Perception Is this Elvis? Sources of Uncertainty Uncertainty about Memory What is this actor’s name? Sources of Uncertainty Uncertainty about Memory Who committed the crime? Sources of Uncertainty Uncertainty about Testimony What will the high temperature be on Friday? On Tuesday? Sources of Uncertainty Uncertainty about Testimony Is the COVID vaccine dangerous? Uncertainty about decisions Uncertainty about Future What’s the fastest route to MOMA from the Lincoln Tunnel? Uncertainty about decisions Uncertainty about Future Will the market go up next year? Probability Probability The probability of an event is a number between 0 (impossible) and 1 (necessary) Subjective probabilities are the probabilities that we assign in our reasoning What subjective probability do you assign to the following events: A fair coin will land heads Taylor Swift will release an album in 2025 Taylor Swift will release a metal album in 2025 Alien abduction is real Aliens live among us Jordan Peterson is an alien Probability Probability(X) = X possibilities Total possibilities [Psst: we’re assuming that possibilities are independent, finite, and equally probable] Probability Probability(roll a 6) = X possibilities 1 = Total possibilities 6 1 2 3 4 5 6 Probability Probability(roll an even number) = X possibilities 3 = Total possibilities 6 1 2 3 4 5 6 Probability Probability(roll number ≥2) = X possibilities 5 = Total possibilities 6 1 2 3 4 5 6 Probability Probability(X | Y) = X-and-Y possibilities Y possibilities Probability Probability(roll 6 | you roll an even number) = X-and-Y possibilities 1 = Y possibilities 3 1 2 3 4 5 6 Probability Probability(roll an even number | roll an even number) = X-and-Y possibilities 3 = Y possibilities 3 1 2 3 4 5 6 Probability Probability(roll number ≥3 | roll an even number) = X-and-Y possibilities 2 = Y possibilities 3 1 2 3 4 5 6 Bayes’s Theorem  Bayes’s Theorem H is a hypothesis, E is your evidence, Pr is the probability assigned to a possibility: Likelihood Pr(H | E) = P(E | H) x P(H) Prior P(E) Normalizing For choice between H and not H: Posterior P(E) = (P(E | H) x P(H))+(P(E | not H) x P(not H)) Constant Bayes’s Theorem H is that you have COVID, E is a positive test, P(H)=.1, P(E | H)=.7, P(E | not H)=.1 P(E | H) x P(H).7 x.1 Pr(H | E) = P(E) = (.7 x.1)+(.1 x.9) =.4375 You should increase your probability from.1 to.4375 after a positive result Bayes’s Theorem H is that you have COVID, E is a positive test, P(H)=.001, P(E | H)=.7, P(E | not H)=.1 P(E | H) x P(H).7 x.001 Pr(H | E) = P(E) = (.7x.001)+(.1x.999) =.007 Consequence 1: If the prior probability of H is sufficiently low, the posterior probability pr(H | E) will be low regardless of E. Basically: If you believe that H is almost impossible, new evidence won’t make much difference. Bayes’s Theorem Consequence 1: If the prior probability of H is sufficiently low, the posterior probability pr(H | E) will be low regardless of E. Basically: If you believe that H is almost impossible, new evidence won’t make much difference. Bayes’s Theorem H is that you have COVID, E is a positive test, P(H)=.1, P(E | H)=.51, and P(E | not H)=.49 P(E | H) x P(H).51 x.1 Pr(H | E) = P(E) = (.51 x.1)+(.49 x.9) =.104 Consequence 2: If the likelihood of Pr(E | H) is about the same as Pr(E | not H), the posterior Pr(H | E) will be close to the prior p(H). Basically: if your evidence E is worthless, so you stick with what you already believed.  “I used to be a ninja, and once Bayes’s Theorem saved Queen Elizabeth’s life… Consequence 2: If the likelihood of Pr(E | H) is about the same as Pr(E | not H), the posterior Pr(H | E) will be close to the prior p(H). Basically: if your evidence E is worthless, so you stick with what you already believed. Bayesian Optimality Bayesian Optimality It will be 60 degrees tomorrow It will be 50 degrees tomorrow 80% reliable weatherman 60% reliable weatherman What temperature is most likely? A weighted average Bayesian Optimality Optimal approach to cue combination: For discrepant cues, you should take their weighted average, where each is weighted by the reliability of that cue.  Bayesian Optimality For discrepant visual and vestibular cues, monkeys seemed to use the Fetsch et al. 2010 weighted average, where each cue was weighted by its reliability. Bayesian Suboptimality Bayesian Suboptimality Representativeness: When asked the probability that A belongs to a class B, people often rely on the degree to which A is resembles a paradigmatic example of B. Bayesian Suboptimality Representativeness: When asked the probability that A belongs to a class B, people often rely on the degree to which A is resembles a paradigmatic example of B. Failures that result from Representativeness: 1. Base rate neglect “Steve is very shy and withdrawn, invariably helpful, but with little interest in people, or in the world of reality. A meek and tidy soul, he has a need for order and structure, and a passion for detail.” Is it more likely that Steve is an engineer or lawyer? Bayesian Suboptimality Representativeness: When asked the probability that A belongs to a class B, people often rely on the degree to which A is resembles a paradigmatic example of B. Failures that result from Representativeness: 1. Base rate neglect “Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.” Is it more likely that Linda is a bank teller or a feminist bank teller? Bayesian Suboptimality Representativeness: When asked the probability that A belongs to a class B, people often rely on the degree to which A is resembles a paradigmatic example of B. Failures that result from Representativeness: 1. Base rate neglect 2. Insensitivity to sample size Ex: Is a large or a small hospital more likely to have >60% boys? Ex: The ten counties with the lowest cancer rate voted Republican by wide margins Bayesian Suboptimality Representativeness: When asked the probability that A belongs to a class B, people often rely on the degree to which A is resembles a paradigmatic example of B. Failures that result from Representativeness: 1. Base rate neglect 2. Insensitivity to sample size 3. Misconceptions of chance Fair coin flip: HTHTHTTHHHH Gambler’s fallacy: Next flip is more likely to be T than H Bayesian Suboptimality Representativeness: When asked the probability that A belongs to a class B, people often rely on the degree to which A is resembles a paradigmatic example of B. Failures that result from Representativeness: 1. Base rate neglect 2. Insensitivity to sample size 3. Misconceptions of chance 4. Insensitivity to predictability Evaluations of a student-teacher’s performance often = Predictions of a student-teacher’s future performance Bayesian Suboptimality Representativeness: When asked the probability that A belongs to a class B, people often rely on the degree to which A is resembles a paradigmatic example of B. Failures that result from Representativeness: 1. Base rate neglect 2. Insensitivity to sample size 3. Misconceptions of chance 4. Insensitivity to predictability 5. Misconceptions of Regression Bayesian Suboptimality Availability: When asked the probability of A, people often rely on the ease with which instances of A can be brought to mind Failures that result from Availability: 1. Influence of familiarity 2. Influence of ease of search 3. Influence of salience 4. Influence of recency 5. Bias of imaginability 6. Illusory correlation Metacognition Metacognition SAT Question Should you answer or leave blank? Metacognition Important results about human metacognition: Confidence often varies independently of accuracy Overconfidence tends to apply across domains More information can lead to overconfidence Expertise can lead to overconfidence We’re bad at predicting which problems we can solve Janet Metcalfe Lisa Son Metacognition Three ways of testing metacognition Kepecs and Mainen 2012 Metacognition Kepecs 2012  Metacognition But maybe they’re just uncertain about the stimulus? Kepecs 2012