HPSLec72010 - Lecture#7 Induction Continued AGENDA I Probabilistic Induction A Reverend Thomas Bayes B Bayes Theorem and Induction C Probabilistic

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1 Lecture #7 Induction Continued AGENDA I. Probabilistic Induction A. Reverend Thomas Bayes B. Bayes Theorem and Induction C. Probabilistic Induction Brain Teasers I. Logical Induction A. Mathematical Induction B. A Brain Teaser I. Human Inductive Heuristics II. Paradoxes of Induction
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2 I. Probabilistic Induction Probability theory developed after plane geometry (Euclid, ca 300 B.C.); analytic geometry (Rene Descartes, 1596-1650); calculus (Sir Isaac Newton, 1642-1727; Gottfried Leibniz) Pascal (1623-1662), Fermat (1601-1665), and Jacob Bernoulli (1654-1705) were among the pioneers. Their results came in part from assisting gamblers. Fermat 1601-1665 Jacob Bernoulli 1654-1705 Pascal 1623-1662
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3 Pascal and Fermat’s problem Suppose a pair of dice is rolled until a boxcar occurs (boxcar= double six). How many rolls are needed to bring the probability over ½ that a boxcar will occur? Well the probability that a boxcar will occur in n tosses is Well for n=25, Pr(boxcar)=0.505, about an equal bet. n - = 36 35 1 rolls) n in boxcar one least at Pr( Hint: Probability at least one head in 3 flips= 1- prob (no heads in three flips)=1-(1/2) 3 = 7/8
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4 A. Reverend Thomas Bayes (1702-1761) Bayes formulated a way to inductively update the strength of beliefs with evidence. Beliefs are represented by probabilities and evidence changes these probabilities. His formula involved conditional probability. Belief (A given evidence E)= (Likelihood of E) = Prior Belief in A Evidence E Concerning A Posterior Belief in A given evidence (Likelihood of E given belief A)(Prior belief A) (Likelihood of E) (Likelihood of E given A)(prior belief A) +(likelihood E given not A)(prior belief not A)
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5 Example of Bayesian Inference Suppose there are two urns: U I has 2 red balls and 1 blue ball, U II has 1 red and 3 blue. A games master selects an urn at random and takes a ball at random . The ball is blue . What is your belief in the probability it came from U I ? Prior : Pr(U I )=1/2. Evidence : Ball is blue Posterior : Pr(U I blue)=? I II
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6 Simple Probability Facts From Statistics If A and B are non-zero probability events, i.e. Then the conditional probability of B given A is defined as From which we have Bayes theorem : 0 ) Pr( ), Pr( B A ) Pr( ) Pr( ) Pr( A B A A B = ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( ) Pr( A B B A A B A A B = = Prob both A and B occur
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7 The Solution Using Bayes Theorem The probability of the evidence is obtained by simple probability theory: (1/3)(1/2)+(3/4)(1/2)=13/24 13 / 4 ) 24 / 13 ( ) 2 / 1 ( ) 3 / 1 ( (B) Pr ) U Pr( ) U Pr(B B) U Pr( I I I = = = Posterior belief probability Probability of evidence given U I Probability of evidence Prior belief probability = + = ) U Pr( ) U B Pr( ) U Pr( ) U Pr(B B) Pr( II II I I
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8 B. Bayes Theorem and Induction Bayes thought of H as a scientific hypothesis and E as the evidence. It is often easy to calculate the probability of the evidence given the hypothesis.
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This note was uploaded on 12/05/2010 for the course PSYCH Psy Beh F2 taught by Professor Williamh.batchelder during the Fall '10 term at UC Irvine.

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HPSLec72010 - Lecture#7 Induction Continued AGENDA I Probabilistic Induction A Reverend Thomas Bayes B Bayes Theorem and Induction C Probabilistic

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