lecture11-2k - AEB 6182 Lecture XI Professor Charles Moss...

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AEB 6182, Lecture XI Professor Charles Moss 1 Value of Information Lecture XI I. Decision Making and Bayesian Probabilities A. Traditionally, Bayesian analysis involves a procedure whereby new information is integrated into a prior distribution to generate an updated or posterior distribution. B. At times the concept of Bayesian probability theory is confused with the subjective probability theory where an individual has an intuition regarding the probability of an event instead of a frequency view of probability theory which strives for an objective version of probability. C. At the base of Bayesian inference is Bayes’s eqaution: P a b P a b P b [ | ] [ , ] [ ] = where P[a|b] is the probability of the event a occurring such that event a has already occurred, P[a,b] is the joint probability of both event a and event b occurring, and P[b] is the marginal probability of event b occurring (or the probability of event b occurring such that event a has been integrated from the distribution). D. This basic concept of prior and posterior probabilities can be used to develop one manifestation of the value of information. Specifically, imagine the state space: E 1 E 2 E 2 O 1 O 2 P[O 1 ]=.5 P[O 2 ]=.5 O 1|1 O 2|1 O 1|2 O 2|2 P[O 1 |O 1 ]=.7 2 |O 1 ]=.3 P[O 1 |O 2 ]=.3 P[O 2 |O 2 ]=.7
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AEB 6182, Lecture XI Professor Charles Moss 2 1. This diagram depicts the potential outcomes of two random events each of which has two potential outcomes. a. The first event yields outcomes O 1 and O 2 each of which occurs with probability .5. b. The second event results in O 1|1 and O 2|1 if O 1 occurred in the first event and O 1|2 and O 2|2 given that O 2 occurred in the first event. c. Intuitively, we can picture O and O as the same event with a different intervening event. Similarly, O and O are the same event. d. What does change with the intervening event is the relative probability that each outcome will occur. i. Given O 1 the probability of outcome 1 in the second stage (O ) is .7 compared with a probability of outcome 2 in the second stage (O ) of .3.
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lecture11-2k - AEB 6182 Lecture XI Professor Charles Moss...

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