THE QUASI-BAYESIAN FRAMEWORK

 

Consider this problem: an agent must choose a plan $a$_i before the state of the world is known; after the state is revealed to be $\theta$_j, the agent pays a loss $l$_ij. The losses indicate the preferences of the agent. How should the agent compare two plans, $a$₁ and $a$₂? Quasi-Bayesian theory asserts that there is a nonempty convex set $K$ of probability distributions which summarizes the agent's beliefs. The set $K$ is such that, for plans $a$₁ and $a$₂, $a$₁ is at least as preferred as $a$₂ iff $E[l$_1j] >= E[l_2j] for every probability distribution $K$, where $E[.]$ denotes expected loss. Giron and Rios present a set of axioms that validate the preferences of a Quasi-Bayesian agent [Giron and Rios1980]. The set of probability distributions $K$ is called the credal set [Levi1980]. The representation of preferences conditional on a state is characterized by a convex set of posterior distributions obtained through application of Bayes rule to each one of the distributions in the set of priors.

There are other methods for creating sets of probability distributions: inner and outer measures [Good1983, Halpern and Fagin1992, Ruspini1987, Suppes1974], intervals of probability (commonly generated by lower probability) [Breese and Fertig1991, Chrisman1995, Fine1988, H. E. Kyburg Jr.1987, Halpern and Fagin1992, Smith1961], lower expectations [Walley1991]. The belief functions used in Dempster-Shafer theory [Ruspini1987, Shafer1987] have different interpretations but can be represented as sets of probabilities. Quasi-Bayesian models generalize these ideas. Given a Quasi-Bayesian convex set of probability distributions, a probability interval can be created for every event $A$ by defining lower and upper bounds:

$P*(A)\; =inf$_{P inK} P(A)

$P**(A)\; =sup$_{P inK} P(A)

In a different direction, more general models than the Quasi-Bayesian one can be created, for instance theories of decision which use simultaneous sets of losses and probabilities [Levi1980, Seidenfeld1993].

There are some basic reasons for adopting a Quasi-Bayesian model [Seidenfeld and Wasserman1993]. First, Quasi-Bayesian theory builds a realistic account of the imperfections in an agent's beliefs. It can be used to represent poor elicitation of preferences and situations of indifference among actions. Second, robustness studies can be formalized through this model [Berger1985]. Third, the theory can represent the disparate opinions of a group of agents [Levi1980].