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 . ${\displaystyle \infty }$

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

${\displaystyle \kappa ({\mbox{Kopf}})=\kappa ({\mbox{Zahl}})=0,\kappa ({\mbox{Rand}})=3}$

be modeled.

definition

A rank function is a map ${\displaystyle \kappa }$

${\displaystyle \kappa \colon 2^{W}\to \mathbf {N} ^{*}}$,

in which

${\displaystyle \mathbf {N} ^{*}=\mathbf {N} \cup \{\infty \}}$

from a subset of a set W of possible worlds into infinitely added natural numbers (including 0), with the following properties:

• (Rk 1):

${\displaystyle \kappa (\emptyset )=\infty }$

• (Rk 2):

${\displaystyle \kappa (W)=0}$

• (Rk 3):

${\displaystyle \kappa (U\cup V)=\min(\kappa (U),\kappa (V))}$If and disjoint are ${\displaystyle U}$ ${\displaystyle V}$

So that the rank is determined by the single-element sets ( singletons ) even for infinite sets ,

${\displaystyle \kappa (U)=\min _{u\in U}\kappa (u)}$,

what then requires at least one element from with rank 0 to adhere to (Rk 2) , one demands the tightening ${\displaystyle W}$

• (Rk 3+):

${\displaystyle \kappa (\bigcup _{i\in I}U_{i})=\min\{\kappa (U_{i})|i\in I\}}$for any index sets and pairwise disjoint indexed sets ${\displaystyle I}$ ${\displaystyle U_{i}}$

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). ${\displaystyle W=\mathbf {N} }$ ${\displaystyle \infty }$

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