The Difference by Page Scott E

Author:Page, Scott E. [Page, Scott E.]
Language: eng
Format: epub
ISBN: 978-1-4008-3028-2
Publisher: Princeton University Press
Published: 2007-03-25T16:00:00+00:00

THE DIVERSITY PREDICTION THEOREM

Now that we’ve worked through an example, we’re ready to turn to the more general theorems that reveal the importance of diverse predictive models among the members of a crowd. Versions of these theorems can be found in computer science, statistics, and econometrics.7 To describe these theorems, we’ll need two measures. The first captures how much a collection of predictive models differs. The other captures how accurate the models are. Both are based on the same accuracy measure: squared errors. In statistics, errors are squared so that negative errors and positive errors do not cancel one another out. If errors were added, a person who was equally likely to overestimate or underestimate an amount on average would make no errors (—5 + 5 = 0). If we first square the errors, then the negative and positive errors do not cancel ((—5)2 + 52 = 25 + 25 = 50). To build the logic of the theorem, we first construct an example. Suppose that Micheala and Juliana have developed models to predict where three students—Maggie, Cole, and Brody—will place in an upcoming spelling bee at Rudy Giuliani Elementary. Table 8.8 shows their individual predictions, their average prediction, and the actual outcome from the bee.

We first compute the squared errors of Micheala and Juliana’s predictions. Michaela picks Maggie to take sixth place and she takes sixth, an error of zero. She picks Cole to take third and he takes fifth, an error of two. And she picks Brody to take fifth, but he takes first place, an error of four. Squaring these three errors gives zero, four, and sixteen. The sum of the her errors equals twenty.

Micheala’s Individual Error: (6 − 6)2 + (3 − 5)2 + (5 − 1)2 =0 + 4 + 16 = 20

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