An axiomatic model of non bayesian updating
In the 1980s there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery of Markov chain Monte Carlo methods and the consequent removal of many of the computational problems, and to an increasing interest in nonstandard, complex applications.
The use of Bayesian probabilities as the basis of Bayesian inference has been supported by several arguments, such as Cox axioms, the Dutch book argument, arguments based on decision theory and de Finetti's theorem. Cox showed that Bayesian updating follows from several axioms, including two functional equations and a hypothesis of differentiability.
If time permits, I will also characterize the rate of convergence for this family of updates and describe how the rate depends on the network structure and information endowment of agents.
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.Rather than proposing functional form for the belief updates, I will present behavioral axioms from which updates are derived.The first behavioral assumption is a notion of imperfect recall, according to which agents take the current belief of their neighbors as sufficient statistics, ignoring how and why their current opinions were formed.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.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).
A Dutch book is made when a clever gambler places a set of bets that guarantee a profit, no matter what the outcome of the bets.