# 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