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,^[1]^[2] and popularized in Jeffrey's book The Logic of Decision.^[3]^{[note 1]} 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.^[2] 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.^[4]

In the Bolker-Jeffrey formalization, probabilities and utilities are functions of propositions in a complete, atomless Boolean algebra.^[5] More specifically, propositions are identified with finite lists of restrictions on the state of the world,^[6] where the number of states is required to be uncountably infinite.^[7] 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.^[8]

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.^[9]

Definitions edit

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.^[10]

Axioms edit

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 edit

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 $B$ .^[11] 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.^[12]

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 edit

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

the determinant $ad-bc$ is positive,
$c\;U(X)+d$ is positive for all X in the preference ordering, and
$c\;U(T)+d=1$ .^[13]

AI researcher Abram Demski dubbed this kind of transformation a "Jeffrey-Bolker rotation."^[14]

Noncausality edit

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.^[15] 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.

— The Logic of Decision, (1983), page 157 [emphasis added, reformatted for brevity]

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.^[16]

Extensions edit

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,^[17] 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.^[18] 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.^[19]

Notes edit

^ The first edition of The Logic of Decision was published in 1965, before Bolker's representation theorem was published. The second edition, published in 1983, added a chapter on Bolker's axioms.

References edit

^ Bolker 1966.
^ ^a ^b Bolker 1967.
^ Jeffrey 1983.
^ Briggs 2019, "Notice that fractional linear transformations of a probability-utility pair can disagree with the original pair about which propositions are likelier than which others".
^ Briggs 2019, "Bolker's theorem assumes a single domain of propositions, which are objects of preference, utility, and probability alike".
^ Bolker 1967, p. 335: "We cannot really question our subject about a state of the universe because specifying one requires too much information. "Swimming tomorrow" is a set of states, not a state, for there are infinitely many ways in which it can come about. In some the subject swims alone, in others with company. The water temperature can vary. We should think of s as the event consisting of those states in which he swims. It might be reasonable, then, to use for S not the algebra of all subsets of M but the algebra of those subsets which are described by a finite list of restrictions on the members they contain".
^ Bolker 1967, pp. 335-337: "We must have algebras which are both atom free and complete. The classical example of such an algebra is the algebra of measurable subsets of the unit interval... we can interpret the completeness condition as forcing us to pass from a discrete infinity of states to a continuum, just as repudiating atoms forced us to abandon a finite state space".
^ Hájek 2019, "1. Kolmogorov’s Probability Calculus".
^ Joyce 2000, S3: "Although Jeffrey does not do things this way, it is useful to think of this theory as being divided into three distinct components: a formal axiology, or theory of rational preference, that tells us which preference rankings are rational, but does so without placing any constraints on confidence rankings; an epistemology, or theory of rational belief; that tells us which confidence rankings are rational without placing any constraints on the agent's preferences; and a theory of coherence, that tells us which preference rankings and confidence rankings can be rationally held together".
^ Jeffrey 1983, p. 148.
^ Bolker 1966, p. 336.
^ Jeffrey 1983, p. 146.
^ Jeffrey 1983, p. 97.
^ Demski 2018.
^ Jeffrey 1983, p. 157.
^ Maher 1987, p. 166.
^ Villegas 1964.
^ Briggs 2019, "Joyce (1999) shows that with additional resources, Bolker's theorem can be modified to pin down a unique P, and a U that is unique up to positive linear transformation. We need only supplement the preference ordering with a primitive “more likely than” relation, governed by its own set of axioms, and linked to belief by several additional axioms. Joyce modifies Bolker's result to show that given these additional axioms, the “more likely than” relation is represented by a unique P, and the preference ordering is represented by P together with a utility function that is unique up to positive linear transformation".
^ Joyce 2000, S1, S9: "Current opinion has it that there are two flaws in Jeffrey's theory. First, there is the non-uniqueness problem. In contrast with other decision theories, it is not possible within Jeffrey's framework to secure expected utility representations for preferences that are unique up to the choice of a unit and a zero for measuring utility (except in special circumstances)... Result 2 shows that the non-uniqueness inherent in Jeffrey's theory of preference can be removed by imposing constraints directly on confidence rankings. Non-uniqueness is thus not an intrinsic feature of preferences in Jeffrey's theory; rather, it is a consequence of trying to make a theory of rational preference do something that only can be accomplished when such a theory is combined with an epistemology".

Works cited edit

Bolker, Ethan D. (1966). "Functions resembling quotients of measures" (PDF). Transactions of the American Mathematical Society. 124 (2): 292–312. Retrieved 2 March 2022.
Bolker, Ethan D. (1967). "A Simultaneous Axiomatization of Utility and Subjective Probability" (PDF). Philosophy of Science. 34 (4): 333–340. Retrieved 2 March 2022.
Briggs, Rachael (2019). "Normative Theories of Rational Choice: Expected Utility". The Stanford Encyclopedia of Philosophy. Stanford University. Retrieved 2 March 2022.
Demski, Abram. "Probability is Real, and Value is Complex". LessWrong.
Hájek, Alan (2019). "Interpretations of Probability". The Stanford Encyclopedia of Philosophy. Stanford University. Retrieved 2 March 2022.
Jeffrey, Richard (1983). "9. Existence: Bolker's Axioms". The Logic of Decision (2nd ed.). University of Chicago Press. ISBN 978-0226395821.
Joyce, James M. (2000). "Why we still need the logic of decision" (PDF). Philosophy of science. 67: S1–S13. Retrieved 2 March 2022.
Maher, Patrick (1987). "Causality in the logic of decision" (PDF). Theory and Decision. 22 (2): 155–172.
Villegas, Carlos (1964). "On Qualitative $\sigma$ -algebras" (PDF). The Annals of Mathematical Statistics. 35 (4): 1787-1796. Retrieved 5 April 2022.

[4] The first edition of The Logic of Decision was published in 1965, before Bolker's representation theorem was published. The second edition, published in 1983, added a chapter on Bolker's axioms.

[FOOTNOTEBolker1966-1] Bolker 1966.

[FOOTNOTEBolker1967-2] Bolker 1967.

[FOOTNOTEJeffrey1983-3] Jeffrey 1983.

[FOOTNOTEBriggs2019"Notice_that_fractional_linear_transformations_of_a_probability-utility_pair_can_disagree_with_the_original_pair_about_which_propositions_are_likelier_than_which_others"-5] Briggs 2019, "Notice that fractional linear transformations of a probability-utility pair can disagree with the original pair about which propositions are likelier than which others".

[FOOTNOTEBriggs2019"Bolker's_theorem_assumes_a_single_domain_of_propositions,_which_are_objects_of_preference,_utility,_and_probability_alike"-6] Briggs 2019, "Bolker's theorem assumes a single domain of propositions, which are objects of preference, utility, and probability alike".

[FOOTNOTEBolker1967p._335:_"We_cannot_really_question_our_subject_about_a_state_of_the_universe_because_specifying_one_requires_too_much_information._"Swimming_tomorrow"_is_a_set_of_states,_not_a_state,_for_there_are_infinitely_many_ways_in_which_it_can_come_about._In_some_the_subject_swims_alone,_in_others_with_company._The_water_temperature_can_vary._We_should_think_of_''s''_as_the_event_consisting_of_those_states_in_which_he_swims._It_might_be_reasonable,_then,_to_use_for_'''S'''_not_the_algebra_of_all_subsets_of_'''M'''_but_the_algebra_of_those_subsets_which_are_described_by_a_finite_list_of_restrictions_on_the_members_they_contain"-7] Bolker 1967, p. 335: "We cannot really question our subject about a state of the universe because specifying one requires too much information. "Swimming tomorrow" is a set of states, not a state, for there are infinitely many ways in which it can come about. In some the subject swims alone, in others with company. The water temperature can vary. We should think of s as the event consisting of those states in which he swims. It might be reasonable, then, to use for S not the algebra of all subsets of M but the algebra of those subsets which are described by a finite list of restrictions on the members they contain".

[FOOTNOTEBolker1967pp._335-337:_"We_must_have_algebras_which_are_both_atom_free_and_complete._The_classical_example_of_such_an_algebra_is_the_algebra_of_measurable_subsets_of_the_unit_interval..._we_can_interpret_the_completeness_condition_as_forcing_us_to_pass_from_a_discrete_infinity_of_states_to_a_continuum,_just_as_repudiating_atoms_forced_us_to_abandon_a_finite_state_space"-8] Bolker 1967, pp. 335-337: "We must have algebras which are both atom free and complete. The classical example of such an algebra is the algebra of measurable subsets of the unit interval... we can interpret the completeness condition as forcing us to pass from a discrete infinity of states to a continuum, just as repudiating atoms forced us to abandon a finite state space".

[FOOTNOTEHájek2019"1._Kolmogorov’s_Probability_Calculus"-9] Hájek 2019, "1. Kolmogorov’s Probability Calculus".

[FOOTNOTEJoyce2000S3:_"Although_Jeffrey_does_not_do_things_this_way,_it_is_useful_to_think_of_this_theory_as_being_divided_into_three_distinct_components:_a_formal_axiology,_or_theory_of_rational_preference,_that_tells_us_which_preference_rankings_are_rational,_but_does_so_without_placing_any_constraints_on_confidence_rankings;_an_epistemology,_or_theory_of_rational_belief;_that_tells_us_which_confidence_rankings_are_rational_without_placing_any_constraints_on_the_agent's_preferences;_and_a_theory_of_coherence,_that_tells_us_which_preference_rankings_and_confidence_rankings_can_be_rationally_held_together"-10] Joyce 2000, S3: "Although Jeffrey does not do things this way, it is useful to think of this theory as being divided into three distinct components: a formal axiology, or theory of rational preference, that tells us which preference rankings are rational, but does so without placing any constraints on confidence rankings; an epistemology, or theory of rational belief; that tells us which confidence rankings are rational without placing any constraints on the agent's preferences; and a theory of coherence, that tells us which preference rankings and confidence rankings can be rationally held together".

[FOOTNOTEJeffrey1983148-11] Jeffrey 1983, p. 148.

[FOOTNOTEBolker1966336-12] Bolker 1966, p. 336.

[FOOTNOTEJeffrey1983146-13] Jeffrey 1983, p. 146.

[FOOTNOTEJeffrey198397-14] Jeffrey 1983, p. 97.

[FOOTNOTEDemski2018-15] Demski 2018.

[FOOTNOTEJeffrey1983157-16] Jeffrey 1983, p. 157.

[FOOTNOTEMaher1987166-17] Maher 1987, p. 166.

[FOOTNOTEVillegas1964-18] Villegas 1964.

[FOOTNOTEBriggs2019"Joyce_(1999)_shows_that_with_additional_resources,_Bolker's_theorem_can_be_modified_to_pin_down_a_unique_P,_and_a_U_that_is_unique_up_to_positive_linear_transformation._We_need_only_supplement_the_preference_ordering_with_a_primitive_“more_likely_than”_relation,_governed_by_its_own_set_of_axioms,_and_linked_to_belief_by_several_additional_axioms._Joyce_modifies_Bolker's_result_to_show_that_given_these_additional_axioms,_the_“more_likely_than”_relation_is_represented_by_a_unique_P,_and_the_preference_ordering_is_represented_by_P_together_with_a_utility_function_that_is_unique_up_to_positive_linear_transformation"-19] Briggs 2019, "Joyce (1999) shows that with additional resources, Bolker's theorem can be modified to pin down a unique P, and a U that is unique up to positive linear transformation. We need only supplement the preference ordering with a primitive “more likely than” relation, governed by its own set of axioms, and linked to belief by several additional axioms. Joyce modifies Bolker's result to show that given these additional axioms, the “more likely than” relation is represented by a unique P, and the preference ordering is represented by P together with a utility function that is unique up to positive linear transformation".

[FOOTNOTEJoyce2000S1,_S9:_"Current_opinion_has_it_that_there_are_two_flaws_in_Jeffrey's_theory._First,_there_is_the_''non-uniqueness_problem''._In_contrast_with_other_decision_theories,_it_is_not_possible_within_Jeffrey's_framework_to_secure_expected_utility_representations_for_preferences_that_are_''unique''_up_to_the_choice_of_a_unit_and_a_zero_for_measuring_utility_(except_in_special_circumstances)..._Result_2_shows_that_the_non-uniqueness_inherent_in_Jeffrey's_theory_of_preference_can_be_removed_by_imposing_constraints_directly_on_confidence_rankings._Non-uniqueness_is_thus_not_an_''intrinsic''_feature_of_preferences_in_Jeffrey's_theory;_rather,_it_is_a_consequence_of_trying_to_make_a_theory_of_rational_preference_do_something_that_only_can_be_accomplished_when_such_a_theory_is_combined_with_an_epistemology"-20] Joyce 2000, S1, S9: "Current opinion has it that there are two flaws in Jeffrey's theory. First, there is the non-uniqueness problem. In contrast with other decision theories, it is not possible within Jeffrey's framework to secure expected utility representations for preferences that are unique up to the choice of a unit and a zero for measuring utility (except in special circumstances)... Result 2 shows that the non-uniqueness inherent in Jeffrey's theory of preference can be removed by imposing constraints directly on confidence rankings. Non-uniqueness is thus not an intrinsic feature of preferences in Jeffrey's theory; rather, it is a consequence of trying to make a theory of rational preference do something that only can be accomplished when such a theory is combined with an epistemology".

[1]

[2]

[3]

[note 1]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]