What is Bayesian statistics and why everything else is wrong

We use a single example to explain (1), the Likelihood Principle, (2) Bayesian statistics, and (3) why classical statistics cannot be used to compare hypotheses. 1. The Slater School The example and quotes used in this paper come from Annals of Radiation: The Cancer at Slater School by Paul Brodeur in The New Yorker of Dec. 7, 1992. We use the example only to make a point, not a serious analysis. The Slater school is an elementary school in Fresno, California where teachers and staff were “concerned about the presence of two high-voltage transmission lines that ran past the school . . . .” Their concern centered on the “high incidence of cancer at Slater. . . .” To address their concern, Dr. Raymond Neutra of the California Department of Health Services’ Special Epidemiological Studies Program conducted a statistical analysis on the “eight cases of invasive cancer, . . . , the total years of employment of the hundred and forty-five teachers, teachers’ aides, and staff members, . . . , [and] the number of person-years in terms of National Cancer Institute statistics showing the annual rate of invasive cancer in American women between the ages of forty and forty-four — the age group encompassing the average age of the teachers and staff at Slater — [which] enabled him to calculate that 4.2 cases of cancer could have been expected to occur among the Slater teachers and staff members . . . .” For the purpose of our illustration we assume (1) that the 145 employees develop (or not) cancer independently of each other and (2) that the chance of cancer, θ, is the same for each employee. Therefore X , the number of cancers among the 145 employees, has a binomial (145, θ) distribution; we write X ∼ Bin(145, θ). For any integer x between 0 and 145, Pr[X = x|θ] = ( 145 x ) θ(1 − θ). The data turned out to be x = 8. According to Neutra, the expected number of cancers is 4.2. Noting that 4.2/145 ≈ 0.03, we formulate a theory: Theory A: θ = 0.03, which says that the underlying cancer rate at Slater is just like the national average. To address the concern at Slater school we want to compare Theory A to alternatives that would better account for the large number of cancers. To illustrate, we propose three additional theories. All together we have Theory A: θ = 0.03, Theory B: θ = 0.04, Theory C: θ = 0.05, and Theory D: θ = 0.06.