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Unformatted text preview: Bayesian Inference STA 113 Artin Armagan Tuesday, November 10, 2009 Bayes Rule...s! p( y) = p(y )p( )/p(y) Posterior Prior Normalizing Constant: p(y )p( )d Likelihood  Sampling Distribution Reverend Thomas Bayes Tuesday, November 10, 2009 Flipping the Coin One of your friends  whos not necessarily the most reliable one  approaches you with a coin for a betting game. He says he wins every time we observe Heads after a ip. After a hundred ips, you lose all your money. At the end of a hundred ips (n=100), we observe eighty Heads (y=80). Something doesnt look right... You remember that statistics class you took last semester! Who thought youd ever use that stuff. Tuesday, November 10, 2009 Bad Coin? You remember from your statistics class that the number of successes, Y, in an experiment with n Bernoulli trials is Binomially distributed with a probability of success . P(Y=y ,n) = n C k y (1 ) ny It looks like a reverse problem since you know how many successes (Heads) you observed but dont know the underlying proportion ( ) that generated these successes. The data you observed deFnitely suggests a potential value for . You remember that the maximum likelihood estimator for in this case would be 80/100=0.8 which is far from fair. Is this sufFcient information to conclude that this is not a fair coin though? Tuesday, November 10, 2009 Likelihood x Prior You remember that you could make probabilistic statements about using some Bayesian stuff from your statistics class. You have your sampling distribution (Binomial), and need a prior distribution over . Since you dont have much of an idea about what may be a priori, you suggest that it can be any value in [0,1]. To assert your ignorance about , you use a uniform distribution over [0,1] as a prior...
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This note was uploaded on 01/17/2010 for the course STA 113 taught by Professor Staff during the Fall '08 term at Duke.
 Fall '08
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