Cox's theorem

Cox's theorem shows that degrees of plausibility satisfying specified consistency assumptions can be represented by probability measures. It provides an axiomatic route from qualitative reasoning to probability calculus.

Plausibility measure: A numerical representation of how plausible a proposition is given available information. Cox's theorem begins with plausibility values before deriving probability rules.
Cox's product rule: A rule expressing the plausibility of a conjunction through conditional plausibilities. It captures how Cox's consistency assumptions constrain conjunction.
Dutch book argument: An argument that incoherent betting odds expose a bettor to a guaranteed loss. It motivates probability coherence through betting commitments rather than Cox's plausibility assumptions.
Richard Threlkeld Cox: An American physicist who formulated a theorem deriving probability rules from plausibility consistency. His 1946 book presented the result now known by his name.
Bayesian inference: Inference that updates probability distributions using evidence and Bayes' theorem. Cox's result is often used to justify probability as the calculus of consistent inference.
Proposition: A statement that can be assessed as true or false within a logical system. The theorem assigns plausibility to propositions and their combinations.
Cox's sum rule: A rule relating the plausibility of a negation to the plausibility of its proposition. It captures how consistency constrains negation and disjoint alternatives.
de Finetti's theorem: A representation theorem connecting exchangeable sequences of binary outcomes with mixtures of independent Bernoulli trials. It derives probabilistic structure from exchangeability, not Cox's consistency conditions.
Edwin T. Jaynes: An American physicist who developed Bayesian probability as an extended logic of plausible inference. Jaynes made Cox's theorem a central justification for probability as logic.
Bayesian networks: Directed acyclic graphs encoding conditional dependencies among random variables. Their probability calculations rely on the conditional and product rules Cox helps motivate.

Linked from 1 page