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Unformatted text preview: Probability and Probability and Bayes Bayes ’ ’ Theorem Theorem Statistical Viewpoints Statistical Viewpoints • Frequentist – Variation in the estimates reflects what one would observe under repeated sampling – Reflecting information about a single true, but unknown mean • Bayesian – Variation represents probability of occurrence – The mean may vary and there may be some additional information we can bring to bear on estimating it Probability and Random Probability and Random Variables Variables random. is and fixed is that Note ) , , ( : notation code our In ) 1 ( ) ( Then, ) , ( ~ say that s Let' X x p n x pbinom p p i n x X P p n Bin X i n i x i ¡ ¡ ¸ ¸ ¹ · ¨ ¨ © § d ¦ Probability that the random Probability that the random variable is between a and b variable is between a and b • The probability that X will be between a and b. fixed. are b and a random, is again that Note ) , , ( ) , , ( : notation code our In ) 1 ( ) ( Then, ) , ( ~ Let X p n a pbinom p n b pbinom p p i n b X a P p n Bin X i n i b a i ¡ ¡ ¸ ¸ ¹ · ¨ ¨ © § d d ¡ ¦ Confidence Interval Confidence Interval • Note that Xbar is the random variable. • What we want to say is “The probability that P is between a and b is 95%.” • What we actually must say is “The interval will contain the true mean P under repeated sampling 95% of the time.” ¸ ¹ · ¨ © § ¡ ¢ ¢ £ ¸ ¸ ¹ · ¨ ¨ © § ¢ £ ¢ £ n X n X P n X P V P V V P 96 . 1 96 . 1 95 . 96 . 1 / 96 . 1 Bayesian Paradigm Bayesian Paradigm • But, what if we could treat P as a random variable? • Then we might be able to say P(a< P <b) or P( P <b) • It turns out we might do this using what we already know about conditional probability. • But to do so, we also need to make some additional assumptions about the nature of P¡ Bayesians vs. Bayesians vs. Non Non Bayesians Bayesians • A frequentist uses impeccable logic to answer the wrong question, while a Bayesian answers the right question by making assumptions that nobody can fully believe in. – P. G. Hammer Overview Overview • Today we will explore the Bayesian viewpoint and see how we can bring additional information into the estimation process • But first a bit of a review of probability. P(Heart)=? P(Heart)=4/16 =1/4 Events of Number Total Events of Number occurs event an y that Probabilit ) ( A A A P Probability of an Event Probability of an Event P(Heart,Red)= 2/16=1/8 P(HeartRed)=? P(HeartRed)= P(Heart,Red)/ P( Red) P(HeartRed)= 2/16 / 1/4 = 1/ 2 occurs y that Probabilit occur and both y that Probabilit ) ( ) , ( occurs given that occurs event an y that Probabilit )  ( B B A B P B A P B A B A P ) ( )  ( ) , ( implies ) ( ) , ( )  ( that Note B P B A P B A P B P B A P B A P ) ( ) , ( )  ( A P B A P A B P Probability of B given A Probability of B given A )...
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This note was uploaded on 11/29/2010 for the course STSCI 3010 at Cornell.
 '10
 SULLIVAN

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