Conditional event algebra
Encyclopedia
A conditional event algebra (CEA) is an algebraic structure
whose domain consists of logical objects described by statements of forms such as "If A, then B," "B, given A," and "B, in case A." Unlike the standard Boolean algebra of events, a CEA allows the defining of a probability function, P, which satisfies the equation P(If A then B) = P(A and B) / P(A) over a usefully broad range of conditions.
, one begins with a set, Ω, of outcomes (or, in alternate terminology, a set of possible worlds
) and a set, F, of some (not necessarily all) subsets of Ω, such that F is closed under the countably infinite versions of the operations of basic set theory: union (∪), intersection (∩), and complementation ( ′). A member of F is called an event (or, alternatively, a proposition
), and F, the set of events, is the domain of the algebra. Ω is, necessarily, a member of F, namely the trivial event "Some outcome occurs."
A probability function P assigns to each member of F a real number, in such a way as to satisfy the following axioms
:
It follows that P(E) is always less than or equal to 1. The probability function is the basis for statements like P(A ∩ B′) = 0.73, which means, "The probability that A but not B is 73%."
P(B|A) equals, by definition, P(A ∩ B) / P(A).
It is tempting to write, instead, P(A → B) = 0.24, where A → B is the conditional event "If A, then B." That is, given events A and B, one might posit an event, A → B, such that P(A → B) could be counted on to equal P(B|A). One benefit of being able to refer to conditional events would be the opportunity to nest conditional event descriptions within larger constructions. Then, for instance, one could write P(A ∪ (B → C)) = 0.51, meaning, "The probability that either A, or else if B, then C, is 51%."
Unfortunately, philosopher David Lewis
showed that in orthodox probability theory, only certain trivial Boolean algebras with very few elements contain, for any given A and B, an event X which satisfies P(X) = P(B|A). Later extended by others, this result stands as a major obstacle to any talk about logical objects that can be the bearers of conditional probabilities.
Conditional event algebras circumvent the obstacle identified by Lewis by using a nonstandard domain of objects. Instead of being members of a set F of subsets of some universe set Ω, the canonical objects are normally higher-level constructions of members of F. The most natural construction, and historically the first, uses ordered pairs of members of F. Other constructions use sets of members of F or infinite sequences of members of F.
Specific types of CEA include the following (listed in order of discovery):
CEAs differ in their formal properties, so that they cannot be considered a single, axiomatically characterized class of algebra. Goodman-Nguyen-van Frassen algebras, for example, are Boolean while Calabrese algebras are non-distributive
. The latter, however, support the intuitively appealing identity A → (B → C) = (A ∩ B) → C, while the former do not.
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
whose domain consists of logical objects described by statements of forms such as "If A, then B," "B, given A," and "B, in case A." Unlike the standard Boolean algebra of events, a CEA allows the defining of a probability function, P, which satisfies the equation P(If A then B) = P(A and B) / P(A) over a usefully broad range of conditions.
Standard probability theory
In standard probability theoryProbability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, one begins with a set, Ω, of outcomes (or, in alternate terminology, a set of possible worlds
Possible Worlds
Possible Worlds may refer to:* Possible worlds, a concept in philosophy* Possible Worlds , by John Mighton** Possible Worlds , by Robert Lepage, based on the Mighton play* Possible Worlds , by Peter Porter...
) and a set, F, of some (not necessarily all) subsets of Ω, such that F is closed under the countably infinite versions of the operations of basic set theory: union (∪), intersection (∩), and complementation ( ′). A member of F is called an event (or, alternatively, a proposition
Propositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
), and F, the set of events, is the domain of the algebra. Ω is, necessarily, a member of F, namely the trivial event "Some outcome occurs."
A probability function P assigns to each member of F a real number, in such a way as to satisfy the following axioms
Probability axioms
In probability theory, the probability P of some event E, denoted P, is usually defined in such a way that P satisfies the Kolmogorov axioms, named after Andrey Kolmogorov, which are described below....
:
- For any event E, P(E) ≥ 0.
- P(Ω) = 1
- For any countable sequence E1, E2, ... of pairwise disjoint events, P(E1 ∪ E2 ∪ ...) = P(E1) + P(E2) + ....
It follows that P(E) is always less than or equal to 1. The probability function is the basis for statements like P(A ∩ B′) = 0.73, which means, "The probability that A but not B is 73%."
Conditional probabilities and probabilities of conditionals
The statement "The probability that if A, then B, is 24%" means (put intuitively) that event B occurs in 24% of the outcomes where event A occurs. The standard formal expression of this is P(B|A) = 0.24, where the conditional probabilityConditional probability
In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...
P(B|A) equals, by definition, P(A ∩ B) / P(A).
It is tempting to write, instead, P(A → B) = 0.24, where A → B is the conditional event "If A, then B." That is, given events A and B, one might posit an event, A → B, such that P(A → B) could be counted on to equal P(B|A). One benefit of being able to refer to conditional events would be the opportunity to nest conditional event descriptions within larger constructions. Then, for instance, one could write P(A ∪ (B → C)) = 0.51, meaning, "The probability that either A, or else if B, then C, is 51%."
Unfortunately, philosopher David Lewis
David Kellogg Lewis
David Kellogg Lewis was an American philosopher. Lewis taught briefly at UCLA and then at Princeton from 1970 until his death. He is also closely associated with Australia, whose philosophical community he visited almost annually for more than thirty years...
showed that in orthodox probability theory, only certain trivial Boolean algebras with very few elements contain, for any given A and B, an event X which satisfies P(X) = P(B|A). Later extended by others, this result stands as a major obstacle to any talk about logical objects that can be the bearers of conditional probabilities.
The construction of conditional event algebras
The classification of an algebra makes no reference to the nature of the objects in the domain, being entirely a matter of the formal behavior of the operations on the domain. However, investigation of the properties of an algebra often proceeds by assuming the objects to have a particular character. Thus, the canonical Boolean algebra is, as described above, an algebra of subsets of a universe set. What Lewis in effect showed is what can and cannot be done with an algebra whose members behave like members of such a set of subsets.Conditional event algebras circumvent the obstacle identified by Lewis by using a nonstandard domain of objects. Instead of being members of a set F of subsets of some universe set Ω, the canonical objects are normally higher-level constructions of members of F. The most natural construction, and historically the first, uses ordered pairs of members of F. Other constructions use sets of members of F or infinite sequences of members of F.
Specific types of CEA include the following (listed in order of discovery):
- Shay algebras
- Calabrese algebras
- Goodman-Nguyen-van Fraassen algebraGoodman-Nguyen-van Fraassen algebraA Goodman–Nguyen–van Fraassen algebra is a type of conditional event algebra that embeds the standard Boolean algebra of unconditional events in a larger algebra which is itself Boolean...
s - Goodman-Nguyen-Walker algebras
CEAs differ in their formal properties, so that they cannot be considered a single, axiomatically characterized class of algebra. Goodman-Nguyen-van Frassen algebras, for example, are Boolean while Calabrese algebras are non-distributive
Distributive lattice
In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection...
. The latter, however, support the intuitively appealing identity A → (B → C) = (A ∩ B) → C, while the former do not.