# Statistical Default Theory

**See:** Default Theory, Default Theory Expression, Statistical Argument.

## References

### 2012

- http://en.wikipedia.org/wiki/Default_logic#Variants_of_default_logic
- QUOTE: ... Statistical variant: a statistical default is a default with an attached upper bound on its frequency of error; in other words, the default is assumed to be an incorrect inference rule in at most that fraction of times it is applied.

### 2002

- (Wheeler, 2002) ⇒ Gregory R. Wheeler. (2002). “Statistical Default Logic]."
- QUOTE: Given the parameter [math]\displaystyle{ \epsilon }[/math], how should accepted sentences interact in a statistical default extension? In Reiter’s standard formulation, a default extension is deductively closed. But bounds for frequency of error for particular statistical inferences do not necessarily carry over unchanged when chained together to form a statistical argument. Accepting an hypothesis H1 with a conﬁdence 0.95 and accepting another hypothesis H2 independently with a conﬁdence 0.95 doesn’t entail the acceptance of H1 and H2 at 0.95. So, we cannot simply close a statistical default extension under deduction and guarantee that the deductive consequences will have the same ²-bound as a constituent inference.
When we consider making a statistical inference, we foremost consider the frequency bounds on possible errors to which the inference is subject. Likewise for statistical arguments. Since statistical arguments may contain a number of inferences, both deductive and nonmonotonic, we need a mechanism for identifying permissible inference chains within a given ²-bound.

Neither classical logic nor standard default logic restricts the length of permissible sequences of inferences, however. Inference chains suitable for modelling statistical arguments, on the other hand, must be bounded in error. An inference chain that is deductively valid or one that generates an extension for a given default theory may nevertheless fail to be an acceptable statistical argument because of the error bound of the sequence of inference steps does not fall within the designated error bound. Since a conclusion of a statistical argument is acceptable only if the conclusion is the conclusion of an inference chain bounded in error by a preassigned, it is important to track whether a conclusion is a reached by a sequence of inferences bounded in error by ².

- QUOTE: Given the parameter [math]\displaystyle{ \epsilon }[/math], how should accepted sentences interact in a statistical default extension? In Reiter’s standard formulation, a default extension is deductively closed. But bounds for frequency of error for particular statistical inferences do not necessarily carry over unchanged when chained together to form a statistical argument. Accepting an hypothesis H1 with a conﬁdence 0.95 and accepting another hypothesis H2 independently with a conﬁdence 0.95 doesn’t entail the acceptance of H1 and H2 at 0.95. So, we cannot simply close a statistical default extension under deduction and guarantee that the deductive consequences will have the same ²-bound as a constituent inference.