Report

Phil 148 Bayes’s Theorem/Choice Theory You may have noticed: • The previously discussed rules of probability involved each of the logical operators: negation, disjunction, and conjunction, except for conditional. • Bayes’s Theorem is a theorem of conditional probability. • You’ll notice that we are now progressing beyond a priori probability, and into statistical probability. An Example: • Wendy has tested positive for colon cancer. • Colon cancer occurs in .3% of the population (.003 probability) • If a person has colon cancer, there is a 90% chance that they will test positive (.9 probability of a true positive) • If a person does not have colon cancer, then there is a 3% chance that they will test positive (3% chance of a false positive) • Given that Wendy has tested positive, what are her chances of having colon cancer? Answer: • The correct answer is 8.3% • Most people assume that the chances are much better than they really are that Wendy has colon cancer. The reason for this is that they forget that a test must be absurdly specific to give a high probability of having a rare condition. Formal statement of Bayes’s Theorem: BT: Pr(h | e) = ___Pr(h) * Pr(e|h)___ [Pr(h) * Pr(e|h)] + [Pr(~h) * Pr(e|~h)] h = the hypothesis e = the evidence for h Pr(h) = the statistical probability of h Pr(e|h) = the true positive rate of e as evidence for h Pr(e|~h) = the false positive rate of e as evidence for h The Table Method: e ~e Total h ~h Total True Positives False Negatives Pr(h)*Pop. False Positives True Negatives Pr(~h)*Pop. Pr(e)*Pop. Pr(~e)*Pop. Pop. = 10^n n = sum of decimal places in two most specific probabilities. The Table Method: h e ~e Total ~h = Pr(e|h) * = Pr(e|~h) * [Pr(h)*Pop.] [Pr(~h)*Pop. ] = below = below above above Pr(h)*Pop. Pr(~h)*Pop. Total Total of this row Total of this row Pop. The Table Method for Wendy: h e ~e Total ~h = Pr(e|h) * = Pr(e|~h) * [Pr(h)*Pop.] [Pr(~h)*Pop. ] = below = below above above Pr(h)*Pop. Pr(~h)*Pop. Total Total of this row Total of this row Pop. The Table Method for Wendy: has CC e ~e Total ~ have CC = Pr(e|h) * = Pr(e|~h) * [Pr(h)*Pop.] [Pr(~h)*Pop. ] = below = below above above Pr(h)*Pop. Pr(~h)*Pop. Total Total of this row Total of this row Pop. The Table Method for Wendy: has CC tests positive ~ test positive Total ~ have CC = Pr(e|h) * = Pr(e|~h) * [Pr(h)*Pop.] [Pr(~h)*Pop. ] = below = below above above Pr(h)*Pop. Pr(~h)*Pop. Total Total of this row Total of this row Pop. The Table Method for Wendy: has CC tests positive ~ test positive Total ~ have CC = Pr(e|h) * = Pr(e|~h) * [Pr(h)*Pop.] [Pr(~h)*Pop. ] = below = below above above .003*Pop. .997*Pop. Total Total of this row Total of this row 100,000 The Table Method for Wendy: has CC tests positive ~ test positive Total ~ have CC = Pr(e|h) * = Pr(e|~h) * [Pr(h)*Pop.] [Pr(~h)*Pop. ] = below = below above above 300 99,700 Total Total of this row Total of this row 100,000 The Table Method for Wendy: tests positive ~ test positive Total has CC ~ have CC Total = True Positive Rate (.9) * 300 = below above 300 = Pr(e|~h) * [Pr(~h)*Pop. ] = below above 99,700 Total of this row Total of this row 100,000 The Table Method for Wendy: has CC ~ have CC Total tests positive 270 Total of this row ~ test positive Total = below above 300 = Pr(e|~h) * [Pr(~h)*Pop. ] = below above 99,700 Total of this row 100,000 The Table Method for Wendy: has CC ~ have CC Total tests positive 270 Total of this row ~ test positive Total 30 = Pr(e|~h) * [Pr(~h)*Pop. ] = below above 99,700 300 Total of this row 100,000 The Table Method for Wendy: has CC ~ have CC Total tests positive 270 Total of this row ~ test positive Total 30 = False positive rate (.03) * 99,700 = below above 99,700 300 Total of this row 100,000 The Table Method for Wendy: tests positive ~ test positive Total has CC ~ have CC Total 270 2,991 30 = below above 99,700 Total of this row Total of this row 100,000 300 The Table Method for Wendy: tests positive ~ test positive Total has CC ~ have CC Total 270 2,991 30 96,709 300 99,700 Total of this row Total of this row 100,000 The Table Method for Wendy: tests positive ~ test positive Total has CC ~ have CC Total 270 2,991 3,261 30 96,709 96,739 300 99,700 100,000 The Table Method for Wendy: tests positive ~ test positive Total has CC ~ have CC Total 270 (true positive) 30 (false negative) 300 2,991 (false 3,261 positive) 96,709 (true 96,739 negative) 99,700 100,000 What are Wendy’s chances? tests positive has CC 270 (true positive) ~ have CC 2,991 (false positive) Total 3,261 •Wendy’s Chances are the true positives divided by the number of total tests. That is, 270/3261, which is .083 (8.3%). •Those who misestimate that probability forget that colon cancer is rarer than a false positive on a test. How about a second test? • Note that testing positive (given the test accuracy specified) raises one’s chances of having the condition from .003(the base rate) to .083. • If we use .083 as the new base rate, those who again test positive then have a 73.1% chance of having the condition. • A third positive test (with .731 as the new base rate) raises the chance of having the condition to 98.8% Another example: • I highly recommend reading the discussion question that runs from p.299-302. • See also this excellent Wikipedia write-up that contains an update to the Sally Clark case: http://en.wikipedia.org/wiki/Prosecutor's_fallac y Choice Theory: • The relationship between probability and action is often complex, however we can use simple mathematical operations (so far all we’ve used have been the four arithmetic operations) to assist in making good choices. • The first principles we will look at are: Expected Monetary Value and Expected Overall Value. Expected Monetary Value: • EMV = [Pr(winning) * net gain ($)] – [Pr(losing) * net loss ($)] • Example, Lottery: • EMV = [(1/20,000,000) * $9,999,999] – [(19,999,999/20,000,000) * $1] • That comes out to -$0.50 • That means that you lose 50 cents on the dollar you invest; this is a bad bet. Expected Monetary Value: • Consider an example with twice the odds of winning and twice the jackpot: • Example 2, Lottery: • EMV = [(1/10,000,000) * $19,999,999] – [(9,999,999/10,000,000) * $1] • That comes out to $1 • That means that you gain a dollar for every dollar you invest; this is a good bet. Expected Overall Value • Monetary value is not the only kind of value. This is because money is not an intrinsic value, but only extrinsic. It is only valuable for what it can be exchanged for. • If the fun of fantasizing about winning is worth losing 50 cents on the dollar, then the overall value of the ticket justifies its purchase. • In general, gamblers always lose money. If viewed as a form of entertainment that is worth the expenditure, it has a good value. If people lose more than they can afford, or if the loss hurts them, it has negative value. Diminishing marginal value: • This is a concept that affects expected overall value. • Diminishing marginal value occurs when an increase in something becomes less valuable per increment of increase. • Examples: sleep, hamburgers, shoes, even money (for discussion, how does diminishing marginal value affect tax policy?) Decisions under risk: • When a person has an idea of what different potential outcomes are, but does not know what the chances of such outcomes are, there are a number of strategies that can guide a decision. • Consider the following table: Outcomes (1-4) given choice (A-C) 1 2 3 4 A 11 3 3 3 B 5 5 5 5 C 6 6 6 3 •Dominance is when one choice is as good or better in every outcome as any competing choice. •There is no dominant choice in the above. •If we do not know the probabilities of ourcomes 1-4, we may assume they are equally probable to generate an expected utility. The EU of A and B are equal, at 5. C comes out slightly better at 5.25. •Other strategies that make sense are: •Maximax: Choose the strategy with the best maximum (in this case, A) •Maximin: Choose the strategy with the best minimum (in this case, B) •Which strategy choice makes most sense depends on how risk-averse the situation is. Ch. 12, Exercise III: 1. EMV = [Pr(winning) * net gain ($)] – [Pr(losing) * net loss ($)] That is: EMV = [.9 * $10 ] – [.1 * $10] = $8 This is a good bet, but would you be willing to risk your friend’s life on it? I should say not. So the EMV is positive, but the EOV is not. In other words, the stakes are SO high for failure that it makes sense to use a maximin strategy, which is not to bet. 2. Your own example?