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Unformatted text preview: Chapter 4 Conditional Probability and the Prediction Problem Conditional probability and conditional expectation are two fundamental concepts in stochastic modeling. They are the key causality concepts used in models of evolving dynamical system. For a randomly evolving system, if we wish to model the fact that event A causes B , this is described by requiring that P { B  A } is increased over P { B } . We’ll see both a calculusbased description of conditional probability and a more advanced formulation. The calculus method is suited to conditioning on a finite set of r.v.s, whereas the advanced method is suited to conditioning on an infinite set of r.v.s. We will specifically look at the prediction problem. as an application since conditional probability and prediction are intimately linked concepts. Two particular examples covered in great detail are: the Best mean square prediction , where we’ll see how conditional expectation yields the best mean square predic tor; and Affine prediction 4.1 Conditional Probability As mentioned above, conditional probability is fundamental to modeling causality in a stochastic setting. Figure 4.1 helps illustrate the computation of conditional probability. Note that conditioning on B essentially restricts the sample space to B . From the picture, computing the probability P { A  B } should be related to P { A ∩ B } , and so P {· B } ∝ P {· ∩ B } . (4.1) Because we have P { B  B } = 1, the proportionality constant must be P { B } . Thus, P { A  B } = P { A ∩ B } P { B } . (4.2) If we rewrite the above equation, we also get the expression P { A ∩ B } = P { B } P { A  B } . (4.3) This equation is sometimes called the multiplication rule. 63 Ω A B Figure 4.1: Condtional Probability Example 4.1: In this example, we’ll look at how conditional probability can be used to evaluate a medical test. Suppose there are 60,000 people in the US population carrying a particular virus. We wish to evaluate the performance of a new diagnostic test for this virus. Lab tests have shown the following probabilities for the diagnostic test: Positive Test Negative Test Virus present 0.99 0.01 Virus absent 0.02 0.98 For an ideal test, the above matrix would be the identity. To evaluate the test, we wish to compute P { virus  positive } . This probability will tell us how likely it is that a person who tests positive under the proposed diagnostic test actually has the virus. Given the cost of treatment and additional testing for individuals who test positive, this probability is of central interest to health professionals. To compute this probability, note that the table provides conditional probabilities in which the conditioning variable is the disease state of the patient. We wish to compute a conditional probability in which the conditioning variable is the diagnostic test outcome....
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 Spring '08
 PETERGLYNN
 Conditional Probability, Normal Distribution, Probability theory, Conditional expectation, Hilbert Space Projection Theorem, mean square predictor

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