Rich Cooper wrote:
Hi Azamat,
But there is
more. We have definitions
for four kinds of sets - fuzzy, probabilistic, rough, and crisp sets.
How many
more KINDS of definitions (i.e., aspects like fuzzy, probabilistic, ..)
are
there in principal?
I.e., with
those four examples, is there a
way to describe the “signature” of how various aspects of reality
are chosen to model sets? If so, can the number four (4) be
increased? What
limit could be placed on the increase, if any?
Following: COLYVAN, M. 2008. Is Probability the Only Coherent Approach
to Uncertainty? Risk Analysis, 28(3), pp.645-652.
Kinds of Uncertainty
1. Epistemic Uncertainty: Incomplete knowledge about some determinate
fact.
There are a number of different flavors of this type of uncertainty,
but
they are all amenable to probabilistic interpretation.
For example, with rough sets arising from coarsening of a reference
scale,
the probability of a certain element belonging to the rough set has
the pattern
---------------------------------------
| Member of | Probability |
---------------------------------------
| lower approximation | 1 |
---------------------------------------
| boundary set | between 0 and 1 |
---------------------------------------
| complement of | 0 |
| upper approximation | |
---------------------------------------
2. Linguistic Uncertainty
Vagueness- (also called fuzziness) using terms that admit borderline
cases, such as "adult"
Context Dependence - such as "tall"
Ambiguity - using terms with multiple meanings, such as "bank"
Underspecificity - using unwanted generality
("there are rainy days ahead")
I'm no expert on linguistic uncertainty, so I can't verify if these
four cases really cover everything.
But I have some expertise in epistemic uncertainty, and as a Bayesian,
I do agree with the statement above. But I imagine there are some
statisticians who disagree. And the disagreement hinges on the
ambiguity of "probability"; the Bayesian and the frequentist
definitions.
A frequentist would split uncertainty into systemic uncertainty and
bias, and would only agree on a probabilistic interpretation of the
latter.
This is starting to look like an ontology!
Tara
|