Rank function (probability theory)
A rank function is used to represent uncertainty ; it expresses the degree of surprise associated with the occurrence of the event or the degree of belief.
A rank 0 means no surprise , rank 1 a little surprising , rank 2 quite surprising , and so on. The rank means so surprising that it is impossible .
It is an alternative to the conventional representation with the help of probability theory .
example
The toss of a coin could be based on a rank function
be modeled.
definition
A rank function is a map
- ,
in which
from a subset of a set W of possible worlds into infinitely added natural numbers (including 0), with the following properties:
- (Rk 1):
- (Rk 2):
- (Rk 3):
- If and disjoint are
So that the rank is determined by the single-element sets ( singletons ) even for infinite sets ,
- ,
what then requires at least one element from with rank 0 to adhere to (Rk 2) , one demands the tightening
- (Rk 3+):
- for any index sets and pairwise disjoint indexed sets
A counterexample would be for a rank function, which assigns the rank 0 to every infinite subset and the rank to every finite subset . It would satisfy (Rk 1) to (Rk 3).
history
Rank functions were first defined by Wolfgang Spohn under the name ordinal conditional functions. You could even take ordinal numbers as values there (ordinal rank function). The interpretation as the degree of surprise comes from GLS Shackle . The name ranking functions comes from Judea Pearl .
literature
- Halpern, Joseph Y .: Reasoning about Uncertainty , The MIT Press (2003) ISBN 0-262-08320-5 (hc) and (2005) ISBN 0-262-58259-7 (pb)
- Spohn, Wolfgang: Ordinal Conditional Functions. A Dynamic Theory of Epistemic States , in WL Harper, B. Skyrms (eds.), Causation in Decision, Belief Change, and Statistics , vol. II, Kluwer, Dordrecht 1988, pp. 105-134 abstract