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.