Bayesian and Frequentist Statistics

Difference between Bayesian and Frequentist Statistics
in One Example

  • In a random poll of 1,000 Americans, 600 approved of the president’s job performance. 
  • Using Bayes Theorem, a Bayesian infers that
    • There’s a 95 percent probability that 60 percent of Americans approve of the president’s performance, with a margin of error of ±3 percentage points.
  • A frequentist regards the Bayesian’s conclusion as statistically improper (at best) or meaningless (at worst).  The argument:
    • A proper statistical probability must be directly testable by repeated experiments.
    • The statement of population parameter can’t be tested in this way, since (unlike samples) there’s only one population.
    • Therefore, the Bayesian’s statement is statistically improper.
  • Frequentists state the matter using the technical terms confidence interval and confidence level, introduced by Jerzy Neyman in 1937:
    • 60 percent of Americans approve of the president’s job performance, with a confidence interval of ±3 percentage points at the 95% confidence level.
  • Eliminating the jargon, the frequentist’s statement amounts to:
    • Given that 60 percent of Americans approve of the president’s performance, there’s a 95 percent probability that in a random sample of 1000 Americans 60 percent will approve of the president’s performance, with a margin of error of  ±3 percentage points.
  • The “95% percent probability” in the frequentist’s statement is about repeatable samples. In the Bayesian’s statement it’s about the population.

Estimation

  • Population Parameters:
    • For Bayesians, a population parameter is a random variable, computable by Bayes Theorem
    • For frequentists, a statistical probability must be testable by repeated experiments. Samples can be repeated, but not populations.
  • View Two Schools of Thought

Scientific Theories