User:Montgolfière/sandbox/Jeffrey-Bolker axioms

Bolker-Jeffrey theory (or Jeffrey-Bolker theory) is a unified mathematical foundation for Bayesian probability and expected utility theory, put forward by Ethan Bolker and Richard Jeffrey in the late 1960s, and popularized in Jeffrey's book The Logic of Decision. It consists of a set of axioms that constrain the preferences of rational agents, and two theorems which assign subjective probabilities and utilities to any set of propositions, provided only a preference ordering over those propositions. The assignment is only unique up to a fractional linear transformation with positive determinant however, meaning that equally valid Bolker-Jeffrey probability assignments for a given preference ordering may disagree about the relative likelihoods of events.

In the Bolker-Jeffrey formalization, probabilities and utilities are functions of propositions in a complete, atomless Boolean algebra. More specifically, propositions are identified with finite lists of restrictions on the state of the world, where the number of states is required to be uncountably infinite. This is in contrast to the classical Kolmogorov formalization of probability theory, which directly assigns probabilities to (possibly finite) sets of outcomes or states of the world.

Richard Jeffrey espoused this mathematical framework alongside his philosophy of radical probabilism and version of evidential decision theory, but Bolker-Jeffrey theory can be used in combination with other decision theories and epistemologies as well.

Definitions

 * The symbol $$F$$ denotes the impossible proposition, which is assigned probability zero.
 * The supremum, or upper bound, of a set of propositions is a proposition that is implied by every proposition in the set, and which implies every other upper bound on the set.
 * The infimum, or lower bound, of a set of propositions is a proposition that implies by every proposition in the set, and is implied by ever other lower bound of the set.
 * A Boolean algebra is complete if and only if every set of propositions has both a supremum and an infimum.
 * An atom is a proposition $$P$$ other than $$F$$ that is implied by itself and $$F$$, but by no other proposition.
 * An atomless Boolean algebra is one which contains no atoms in the above sense.

Axioms

 * Axiom 1 (Nonnegativity)
 * $$P(X) \geq 0$$
 * Axiom 2 (Normalization)
 * The probability of a certainly true statement is 1: $$P(T) = 1$$
 * Axiom 3 (Additivity)
 * Probabilities are additive: if $$X$$ and $$Y$$ are mutually exclusive then $$P(X \lor Y) = P(X) + P(Y)$$
 * Axiom 4 (Desirability Axiom)
 * $$U(X \lor Y) = \frac{P(X)U(X) + P(Y)U(Y)}{P(X) + P(Y)}$$

Existence theorem
The existence theorem states that for any transitive preference ordering $$\succeq$$ over a complete, atomless Boolean algebra less its impossible element $$F$$, there is a probability measure $$P$$ and corresponding signed measure $$J$$ such that the quotient
 * $$U(\alpha) = \frac{J(\alpha)}{P(\alpha)}$$

is a utility function which represents the ordering, in the sense that
 * $$U(A) > U(B)$$

if and only if the agent prefers $$A$$ to $$B$$. This theorem relies on three axioms:


 * Averaging condition
 * If $$A$$ and $$B$$ are mutually exclusive then
 * a) if $$A \succ B$$, then $$A \succ A \lor B \succ B$$, and
 * b) if $$A \sim B$$, then $$A \sim A \lor B \sim B$$.


 * Informally, the third axiom asserts that disjunction is an "averaging" operation. The proposition "A or B," where A and B are mutually exclusive, must lie somewhere in between A and B in the preference ranking.
 * Impartiality
 * Given that $$A$$ and $$B$$ are mutually exclusive and $$A \sim B$$, if $$A \lor C \sim B \lor C$$ for some $$C$$ mutually exclusive with $$A$$ and $$B$$, and not $$C \sim A$$, then $$A \lor C \sim B \lor C$$ for every such $$C$$.
 * Continuity
 * Whenever the supremum (or infimum) of a chain of prospects lies in a preference interval, all members of the chain after (or before) a certain point lie in that interval.

Nonuniqueness
The Bolker-Jeffrey system has the counter-intuitive property that the subjective probabilities that it attributes to agents are not unique, and the utility functions that it assigns are unique in a weaker sense than in traditional von Neumann-Morgenstern utility theory. In particular, for any pair $$P$$, $$U$$ of probability and utility functions representing a preference ordering, a new pair $$P'$$, $$U'$$ can be constructed which also represent the ordering, so long as they take the form
 * $$P'(X) = P(X)\times (c\; U(X) + d)$$
 * $$U'(X) = \frac{a\; U(X) + b}{c\; U(X) + d}$$,

for some constants $$a, b, c, d$$ where
 * 1) the determinant $$ad - bc$$ is positive,
 * 2) $$c\; U(X) + d$$ is positive for all X in the preference ordering, and
 * 3) $$c\; U(T) + d = 1$$.

AI researcher Abram Demski dubbed this kind of transformation a "Jeffrey-Bolker rotation."

Noncausality
Jeffrey argued that the primary virtue of his formulation of expected utility theory is that it is noncausal. This means that Bolker-Jeffrey theory does not require knowledge of how an agent would revise their preference ranking in the face of counterfactual causal relationships; it only requires that the agent be able to rank logical combinations of propositions already in their preference ordering. This places the theory in contrast to Frank Ramsey and Von Neumann–Morgenstern's theories of expected utility, which make fundamental use of the concept of "gambles" or "lotteries" in which arbitrary events are caused by the outcome of a random process.

To make the argument concrete, Jeffrey uses the following example: "Thus, in Ramsey's theory, if the preference ranking contains the consequences $A$ that 'there will be a thermonuclear war next week' and $B$ that 'there will be fine weather next week' and if it contains any gambles on the proposition $C$ that 'this coin lands head up,' then it must also contain the gamble
 * $A$ if $C$, $B$ if not.

It must contain the gamble
 * There will be a thermonuclear war next week if this coin lands head up, and there will be fine weather next week if not.

However, for the agent to consider that this gamble might be in effect would require him so radically to revise his view of the causes of war and weather as to make nonsense of whatever judgment he might offer; and we are no more able than the agent, to say how he would rank such a gamble among the other propositions in his field of preference.

I take it to be the prinicipal virtue of the present theory, that it makes no use of the notion of a gamble or of any other causal notion."

In essence, Jeffrey holds that it is unreasonable to require that rational agents have preferences about gambles which they believe to be causally impossible. Further, he argues that even when an agent does have preferences about impossible gambles, these preferences are not a suitable basis for making inferences about their beliefs, since the supposition that an impossible gamble is possible will necessarily change the agent's other beliefs in important ways.

Extensions
In a 1999 article, James Joyce extended the Bolker-Jeffrey system by positing a primitive confidence ranking $$.\succeq.$$, such that for any two propositions $$A$$ and $$B$$, $$A .\succeq. B$$ if and only if $$A$$ is believed to be more likely than $$B$$. He proves that under weak assumptions, if $$.\succeq.$$ satisfies Bruno de Finetti's axioms of comparative probability and Villegas' continuity condition, then there is a pair of probability and utility functions $$P$$ and $$U$$ representing the agent's confidence and preference rankings such that $$P$$ is unique and $$U$$ is unique up to a positive linear transformation. Joyce argues that his extension to the theory corrects one of its most serious defects— its inability to uniquely specify the strength of probabilities and utilities.