For objectivists, interpreting probability as extension of logic, probability quantifies the reasonable expectation everyone (even a "robot") sharing the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by Cox's theorem.
This paper models an agent in a three-period setting who does not update according to Bayes ' Rule, and who is self-aware and anticipates her updating behavior when formulating plans.
This paper models an agent in a three-period setting who does not update according to Bayes ’ Rule, and who is self-aware and anticipates her updating behavior when formulating plans.
Early Bayesian inference, which used uniform priors following Laplace's principle of insufficient reason, was called "inverse probability" (because it infers backwards from observations to parameters, or from effects to causes).
In the 20th century, the ideas of Laplace developed in two directions, giving rise to objective and subjective currents in Bayesian practice.
The agent is rational in the sense that her dynamic behavior is derived from a stable preference order on a domain of state-contingent menus of acts.
A representation theorem generalizes (the dynamic version of) Anscombe-Aumanns theorem so that both the prior and the way in which it is updated are subjective.
Johann Pfanzagl completed the Theory of Games and Economic Behavior by providing an axiomatization of subjective probability and utility, a task left uncompleted by von Neumann and Oskar Morgenstern: their original theory supposed that all the agents had the same probability distribution, as a convenience. We did not carry this out; it was demonstrated by Pfanzagl ... Ramsey and Savage noted that the individual agent's probability distribution could be objectively studied in experiments.
Pfanzagl's axiomatization was endorsed by Oskar Morgenstern: "Von Neumann and I have anticipated" the question whether probabilities "might, perhaps more typically, be subjective and have stated specifically that in the latter case axioms could be found from which could derive the desired numerical utility together with a number for the probabilities (cf. The role of judgment and disagreement in science has been recognized since Aristotle and even more clearly with Francis Bacon. Recall that the objective methods for falsifying propositions about personal probabilities have been used for a half century, as noted previously.
In that special case, the prior and posterior distributions were Beta distributions and the data came from Bernoulli trials.
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.
The Dutch book argument was proposed by de Finetti; it is based on betting.