Is there a distinction between strong representations and the unique-ness properties that strong representations of probability representations of
(1)type P\leq Q iff Pr(Q)\leq PR(M) often under the name total preorder or weak order or strict weak order.
(2) with the completeness comparability relation taken to mean type
Aleq B or B leq A; where incomparables are taken not to exist
and type 2> which generally goes under the name strict total order or strict trichotomous weak order, whose representation is taken at (1) where =| and completeness property
(1)A>B iff A>B and x=y iff Pr(q)=PR(M)
(2)trichotomy A=B \vee A< B \vee A>B
within the context of a totally/complete qualiative probability order (ie a strong probability representation theorem) ; again no incomparables> Is there a distinction, the latter being stronger in preserving positive definit-ness or is just less ambiguous (prima facie they appear equivalent, although that may depend on the other axioms used) I presume if there is a distinction its at the quantitative level and the latter is stronger with regard to preserving strict monotony in valued representation.
Respresentation by a real valued regular probability function. Available from: https://www.researchgate.net/post/respresentation_by_a_real_valued_regular_probability_function [accessed Jul 7, 2017].
Is there a distinction between strong representations and the unique-ness properties that strong representations of probability representations of
within the context of a totally/complete qualiative probability order (ie a strong probability representation theorem)
and the usage of Strict weak or st
Although they appear equivalent and both are thought to rule out incomparables,events or maximal events
I get the impression that when the representation is in my mind is a real valued probability function that the below construction does have extra force. Is there some ambiguity by what is meant by A less than or equal to B (even though there are non comparables becaues its taken as a primitive and not defined directed as A=B \vee A
Dear Peter, I apologize.
I am just a bit confused by what is meant by equality here, in partial or total orders, as opposed strict weak orders, that are trichotomous, in the sense that in the numerical representation the equalities could be put into equivalence classes. when they say that distinct elements cannot be ranked equal in a strict total order, is this talking about the numerical representation. or in the order, ie the axiom anit symmetry seems to only imply that if x is ranked equal to y then x
Dear Peter I am considering the issue of homeo-morphism of total orders versus strict total orders, in context of probability representation, where >=\equiv = or > is usally defined as true iff A>B or A=B, Is this case generally,. In total orders. I presume that is why embedding for total orders needed for order to generate the a strict part, I presume that it one only has a total order over y iff F(x)>F(y) would even be well defined. I presume that is why they have the bi-conditional in an embedding, as in
https://en.wikipedia.org/wiki/Embedding
I presumed that this would be no different then a strictly monotone function but i was presumably mis-interpreting the domain of interest. In the context of real valued function with real valued domain there would be no real difference but if the domain of an embedding is a non strict totally ordered set as opposed to a real valued function then I presume that the bi-conditional x=>y iff F(x)>=F(y) woudl generate an injective and possibly strictly monotone rank, otherwise if x>=y xneq y and F(x)=F(y) F(x)\geq F(y) which seems alright but it also entails F(x)leq F(y) which entails x, < are not defined. at all. Although the function genreated could by an embedding could be interpreted that way
well a total order i mean a reflexive transitive anti-symmetric relation
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics
- Probability
- Probability Theory