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BayesianEstimation-6

BayesianEstimation-6 - Two Envelopes Revisited The two...

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1 Two Envelopes Revisited The “two envelopes” problem set -up Two envelopes: one contains $X, other contains $2X You select an envelope and open it o Let Y = $ in envelope you selected o Let Z = $ in other envelope Before opening envelope, think either equally good o So, what happened by opening envelope? E[Z | Y] above assumes all values X (where 0 < X < ) are equally likely o Note: there are infinitely many values of X o So, not true probability distribution over X (doesn’t integrate to 1) Y Y Y Z E Y 4 5 2 1 2 2 1 2 ] | [ Subjectivity of Probability Belief about contents of envelopes Since implied distribution over X is not a true probability distribution, what is our distribution over X? o Frequentist : play game infinitely many times and see how often different values come up. o Problem : I only allow you to play the game once Bayesian probability o Have prior belief of distribution for X (or anything for that matter) o Prior belief is a subjective probability By extension, all probabilities are subjective o Allows us to answer question when we have no/limited data E.g., probability a coin you’ve never flipped lands on heads o As we get more data, prior belief is “swamped” by data The Envelope, Please Bayesian : have prior distribution over X, P(X) Let Y = $ in envelope you selected Let Z = $ in other envelope Open your envelope to determine Y If Y > E[Z | Y], keep your envelope, otherwise switch o No inconsistency! Opening envelop provides data to compute P(X | Y) and thereby compute E[Z | Y] Of course, there’s the issue of how you determined your prior distribution over X… o Bayesian: Doesn’t matter how you determined prior, but you must have one (whatever it is) o Imagine if envelope you opened contained $17.51 Revisting Bayes Theorem Bayes Theorem (
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