Frequentist vs. Bayesian inference

In Pragmatic statistics, inference is concerned with understanding a real-world property $\theta$. This can be anything from a causal effect, a classification system for different objects, a regression coefficient, a function, etc.

A point estimate is a single estimate for the true value of $\theta$.

	Frequentist view	Bayesian view
Definition of probability	The long-run frequency of a property $\theta$	Subjective belief in a property $\theta$
What is $\theta$?	An unknown quantity that already exists or is determined in some way	A random variable that we have a prior belief on (represented as a prior distribution $P(\theta )$)
Distribution of the property	The sampling distribution of $\hat{\theta }$, which shows how the estimate of $\theta$ would have looked with a different random sample	The posterior distribution of $\theta$, which shows the belief about $\theta$ given some observed data
A “95% interval” for $\theta$	The confidence interval that contains $\theta$ 95% of the time if many samples are collected	The credible interval that contains $\theta$ with 95% probability