Notes

1. Gerd Gigerenzer, Rationality for Mortals: How People Cope With Uncertainty (Oxford: Oxford University Press; 2008); an analysis of how people understand the prediction of a “30% chance of rain.”

2.

3. Irving J. Good, Good Thinking: The Foundations of Probability and Its Applications (Mineola, NY: Dover; 1983).

4. Kaplan, Bayesian, pp. 292-294.

5. What does the chance of rain in a national weather report refer to? See http:// wxbrad.com/why-a-50-chance-of-rain-usually-means-a- 100-chance-of- confusion/.

6. “Heuristics.” In Chapter 11, I will contrast Gigerenzer's views (see Note 1) of heuristics as useful cognitive tools with those of Daniel Kahneman, Paul Slovic, and Amos Tversky, Judgment Under Uncertainty: Heuristics and Biases (New York: Cambridge University Press; 1982); Daniel Kahneman, 'Thinking, Fast and Slow (New York: Farrar, Strauss and Giroux; 2011) who see them largely as the source of error and bias.

7. Bayesian terminology is somewhat variable. Another way of expressing Bayes' Theorem in words is that the posterior probability is equal to the prevalence of mari­juana use, p(M/A) among opioid users multiplied by the prevalence of marijuana use, p(M), divided by the prevalence opioid addiction, p(A). p(M/A) is also known as the likelihood.

Bayesian methods can cope with far more complex data and models than the simple ones that I've described. Parameter fitting via Bayesian networks is at the heart of analytic methods using Structured Equation Models (or Structural Causal Models) that we'll encounter in Chapter 15.

8. According to the US Census, the US population as of July 1, 2016, was 323,127,513, of which 77.2% were older than age 18, for a total of 249,454,440 adults.

9. Number of US adults addicted to opioids: 2.1 million, according to https://www. drugabuse.gov/ about-nida/legislative-activities/testimony-to-congress/2016/ americas-addiction-to-opioids-heroin-prescription-drug-abuse.

10. CNN poll on the number of US adults who smoke pot. See http://www.cnn.com/ 2016/08/08/health/marijuana-use-doubles-gallup-poll/index.html.

11. Nate Silver; https://fivethirtyeight.com/politics/elections/.

12. http://www.telegraph.co.uk/news/science/large-hadron-collider/9376804/Higgs- boson-Prof-Stephen-Hawking-loses-100-bet.html.

13. https://www.nytimes.com/2015/12/28/opinion/campaign-stops/250000-a-year-is- not-middle-class.html.

14. http://www.wallstreetoasis.com/salary/investment-banking-compensation.

15. Kaplan, Bayesian, p. 286. Kaplan summarizes the views of a number of Bayesians.

16. Technically, the p(D) in the denominator is a normalizing factor; it ensures that the conditional probabilities add up to 1.0. But if we're comparing hypotheses, we don't really care about the sum. Indeed, in some circumstances, the probabilities won't add up to 1.0: they are improper probabilities.

17. Since p(D/H) is also known as the likelihood, the Bayes' factor is sometimes referred to as the likelihood ratio. The greater the likelihood that a given hypothesis predicts the existing data, the more probable is the hypothesis.

18. This is just an example of how the reasoning behind model testing might go; in real examples, any of the terms to the right of the equal sign in Equation 6.4 will affect the posterior odds ratio and therefore influence which model we decide is preferable.

19. In this form the notation for the likelihood is p(y/0); that is, the probability of observing evidence y, given that parameter 9 is true. In some texts, the likelihood is written as L(9/y). In both cases the meaning is the same.

20. For both boys and girls the growth curve from 3 to 10 years is fairly straight: see www.chartsgraphsdiagrams.com/HealthCharts/height-2-20-boys.html; http://www. chartsgraphsdiagrams.com/HealthCharts/height-2-20-girls.html.

21. See Note 6.

22. David Miller, Critical Rationalism: A Restatement and Defense (Chicago: Open Court; 1994), pp. 125-132.

23. Andrew Gelman and Cosma Rohilla Shalizi, “Philosophy and the Practice of Bayesian Statistics,” British Journal of Mathematical Statistics Psychology 66: 8-38, 2013.

24. Ibid.