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Bayesian%20Inference%20-%20113

Bayesian%20Inference%20-%20113 - Bayesian Inference STA 113...

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Bayesian Inference STA 113 Artin Armagan Tuesday, November 10, 2009
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Bayes’ Rule...s! p( θ |y) = p(y| θ )p( θ )/p(y) Posterior Prior Normalizing Constant: p(y| θ )p( θ )d θ Likelihood - Sampling Distribution Reverend Thomas Bayes Tuesday, November 10, 2009
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Flipping the Coin One of your friends - who’s 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 flip. After a hundred flips, you lose all your money. At the end of a hundred flips (n=100), we observe eighty Heads (y=80). Something doesn’t look right... You remember that statistics class you took last semester! Who thought you’d ever use that stuff. Tuesday, November 10, 2009
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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 θ . P(Y=y| θ ,n) = n C k θ y (1- θ ) n-y It looks like a reverse problem since you know how many successes (Heads) you observed but don’t know the underlying proportion ( θ ) that generated these successes. The data you observed definitely suggests a potential value for θ . You remember that the maximum likelihood estimator for θ in this case would be 80/100=0.8 which is far from fair. Is this sufficient information to conclude that this is not a fair coin though? Tuesday, November 10, 2009
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Likelihood x Prior You remember that you could make probabilistic statements about θ using some Bayesian stuff from your statistics class. You have your sampling distribution (Binomial), and need a prior distribution over θ .
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