I’m teaching an introduction to Bayesian statistics. My first thought was to start with Bayes theorem, as many introductions do. But this isn’t the right starting point. Bayes’ theorem is an indispensable tool for Bayesian statistics, but it is not the foundational principle. The foundational principle of Bayesian statistics is the decision to represent uncertainty by probabilities. Unknown parameters have probability distributions that represent the uncertainty in our knowledge of their values.
Once you decide to use probabilities to express parameter uncertainty, you inevitably run into the need for Bayes theorem to work with these probabilities. Bayes theorem is applied constantly in Bayesian statistics, and that is why the field takes its name from the theorem’s author, Reverend Thomas Bayes (1702-1761). But “Bayesian” doesn’t describe Bayesian statistics quite the same way that “Frequentist” described frequentist statistics. The term “frequentist” gets to the heart of how frequentist statistics interprets probability. But “Bayesian” refers to a Bayes theorem, a computational tool for carrying out probability calculations in Bayesian statistics. If frequentist statistics were analogously named, it might be called “Bernoullian statistics” after Jacob Bernoulli’s law of large numbers.
The term “Bayesian” statistics might imply that frequentist statisticians dispute Bayes’ theorem. That is not the case. Bayes’ theorem is a simple mathematical result. What people dispute is the interpretation of the probabilities that Bayesians want to stick into Bayes’ theorem.
I don’t have a better name for Bayesian statistics. Even if I did, the name “Bayesian” is firmly established. It’s certainly easier to say “Bayesian statistics” than to say “that school of statistics that represents uncertainty in unknown parameters by probabilities,” even though the latter is accurate.