The Mathematical Axioms of Quasi-Bayesian Theory

In this section we take Quasi-Bayesian theory as an example of the kinds of axioms that are used to formalize the theory of sets of probabilities. The assumptions of Quasi-Bayesian theory can be formalized from a small set of axioms about preferences. These technical matters are all collected in this section (you can skip it if you're not interested in axioms and such).

To summarize the basic assumptions: the agent chooses an act a_i and receives the consequence (or lottery) l_ij in case state $\theta_j$ obtains. The set of acts A is assumed the space of all real continuous functions (in fact Giron and Rios [11] have an axiom that states that).

It is postulated that the agent has preferences on the acts. If a₂ is at least as preferred as a₁, then $a_1 \leq a_2$. This basic preference relation can be extended to strict preference: a₂ is strictly preferred to a₁ if $a_1 \leq a_2$ and not $a_2 \leq a_1$. Stric preference is indicated a₁ < a₂.

The following rules are imposed on the preference relation [11]:

1.

The preference relation $\leq$ on A is a partial order.

2.

If $\lambda$ is in the interval (0,1] and a₁ < a₂ then

$$\lambda a_1 + (1 - \lambda) a_3 < \lambda a_2 + (1 - \lambda) a_3.$$

In words: if an act is better than another act, then mixing both acts with the same third act cannot change preferences. An identical axiom is used in von Neumann-Morgenstein theory 1.

3.

If a₁ and a₂ are such that $a_1(\theta) > a_2(\theta)$ for every state $\theta$, then a₁ > a₂.

In words: if the consequences of an act are always better than the consequences of another act no matter the state that obtains, then the first act is better than the second act.

4.

If a_i (i in 1, 2, ...) is such that $a_n \rightarrow a$ and c₁ < a_i < c₂ for all i, then c₁ < a < c₂.

In words: if there is a sequence of acts that converges to a particular act, such that some ordering is always respect by all members of the sequence, then the limiting act obeys that preference ordering also.

Given these axioms, Giron and Rios prove that:

Theorem 1 (Giron and Rios)   There exists a unique nonempty convex set K of finitely additive probability measures such that:

$$a_1 \leq a_2 \Leftrightarrow \int_\Phi a_1 dp \leq \int_\Phi a_2 \; dp \mbox{ for every } p \in K.$$

The set K is the credal set [18].

Note the message of this theorem: acts are judged with respect to expected loss, but two acts can only be compared if all the distributions in K ``agree''. If the distributions ``disagree'', then the two acts cannot be compared. Note that this reproduces exactly the behavior of a partial order: some acts are better than others, and some acts simply cannot be compared.

Additional conditions can be imposed on A in order to make the distributions countable additive, but I will try to simplify the discussion by assuming countable additivity without technical details.

A Digression: Other Types of Axioms for Sets of Probability Distributions

There are other methods for creating sets of probability distributions: inner and outer measures [12,15,22,28], intervals of probability [3,2,7,14,15,16,27], lower expectations [30], belief functions in Dempster-Shafer theory [22,26]. Convex sets of probability generalize these models. 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 [18,24].