It was Pierre-Simon Laplace (1749–1827) who introduced a general version of the theorem and used it to approach problems in celestial mechanics, medical statistics, reliability, and jurisprudence.
[13] In the 20th century, the ideas of Laplace developed in two directions, giving rise to objective and subjective currents in Bayesian practice.
[18] While frequentist statistics remains strong (as demonstrated by the fact that much of undergraduate teaching is based on it [19]), Bayesian methods are widely accepted and used, e.g., in the field of machine learning.
Richard T. Cox showed that Bayesian updating follows from several axioms, including two functional equations and a hypothesis of differentiability.
[10][21] The assumption of differentiability or even continuity is controversial; Halpern found a counterexample based on his observation that the Boolean algebra of statements may be finite.
For example, Hacking writes[23][24] "And neither the Dutch book argument, nor any other in the personalist arsenal of proofs of the probability axioms, entails the dynamic assumption.
[30] Following the work on expected utility theory of Ramsey and von Neumann, decision-theorists have accounted for rational behavior using a probability distribution for the agent.
[34][35] The "Ramsey test" for evaluating probability distributions is implementable in theory, and has kept experimental psychologists occupied for a half century.
[37][38]) Modern work on the experimental evaluation of personal probabilities uses the randomization, blinding, and Boolean-decision procedures of the Peirce-Jastrow experiment.
Personal probabilities are problematic for science and for some applications where decision-makers lack the knowledge or time to specify an informed probability-distribution (on which they are prepared to act).
To meet the needs of science and of human limitations, Bayesian statisticians have developed "objective" methods for specifying prior probabilities.
Finding the right method for constructing such "objective" priors (for appropriate classes of regular problems) has been the quest of statistical theorists from Laplace to John Maynard Keynes, Harold Jeffreys, and Edwin Thompson Jaynes.
Indeed, methods for constructing "objective" (alternatively, "default" or "ignorance") priors have been developed by avowed subjective (or "personal") Bayesians like James Berger (Duke University) and José-Miguel Bernardo (Universitat de València), simply because such priors are needed for Bayesian practice, particularly in science.