34 research outputs found
Interpreting, axiomatising and representing coherent choice functions in terms of desirability
Choice functions constitute a simple, direct and very general mathematical
framework for modelling choice under uncertainty. In particular, they are able
to represent the set-valued choices that appear in imprecise-probabilistic
decision making. We provide these choice functions with a clear interpretation
in terms of desirability, use this interpretation to derive a set of basic
coherence axioms, and show that this notion of coherence leads to a
representation in terms of sets of strict preference orders. By imposing
additional properties such as totality, the mixing property and Archimedeanity,
we obtain representation in terms of sets of strict total orders, lexicographic
probability systems, coherent lower previsions or linear previsions.Comment: arXiv admin note: text overlap with arXiv:1806.0104
Interpreting, axiomatising and representing coherent choice functions in terms of desirability
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that appear in imprecise-probabilistic decision making. We provide these choice functions with a clear interpretation in terms of desirability, use this interpretation to derive a set of basic coherence axioms, and show that this notion of coherence leads to a representation in terms of sets of strict preference orders. By imposing additional properties such as totality, the mixing property and Archimedeanity, we obtain representation in terms of sets of strict total orders, lexicographic probability systems, coherent lower previsions or linear previsions
Coherent and Archimedean choice in general Banach spaces
I introduce and study a new notion of Archimedeanity for binary and
non-binary choice between options that live in an abstract Banach space,
through a very general class of choice models, called sets of desirable option
sets. In order to be able to bring an important diversity of contexts into the
fold, amongst which choice between horse lottery options, I pay special
attention to the case where these linear spaces don't include all `constant'
options.I consider the frameworks of conservative inference associated with
Archimedean (and coherent) choice models, and also pay quite a lot of attention
to representation of general (non-binary) choice models in terms of the
simpler, binary ones.The representation theorems proved here provide an
axiomatic characterisation for, amongst many other choice methods, Levi's
E-admissibility and Walley-Sen maximality.Comment: 34 pages, 7 figure
A theory of desirable things
Inspired by the theory of desirable gambles that is used to model uncertainty
in the field of imprecise probabilities, I present a theory of desirable
things. Its aim is to model a subject's beliefs about which things are
desirable. What the things are is not important, nor is what it means for them
to be desirable. It can be applied to gambles, calling them desirable if a
subject accepts them, but it can just as well be applied to pizzas, calling
them desirable if my friend Arthur likes to eat them. Other useful examples of
things one might apply this theory to are propositions, horse lotteries, or
preferences between any of the above. Regardless of the particular things that
are considered, inference rules are imposed by means of an abstract closure
operator, and models that adhere to these rules are called coherent. I consider
two types of models, each of which can capture a subject's beliefs about which
things are desirable: sets of desirable things and sets of desirable sets of
things. A crucial result is that the latter type can be represented by a set of
the former
Computable randomness is about more than probabilities
We introduce a notion of computable randomness for infinite sequences that
generalises the classical version in two important ways. First, our definition
of computable randomness is associated with imprecise probability models, in
the sense that we consider lower expectations (or sets of probabilities)
instead of classical 'precise' probabilities. Secondly, instead of binary
sequences, we consider sequences whose elements take values in some finite
sample space. Interestingly, we find that every sequence is computably random
with respect to at least one lower expectation, and that lower expectations
that are more informative have fewer computably random sequences. This leads to
the intriguing question whether every sequence is computably random with
respect to a unique most informative lower expectation. We study this question
in some detail and provide a partial answer
A Gentle Approach to Imprecise Probabilities
The field of of imprecise probability has matured, in no small part because of Teddy Seidenfeld’s decades of original scholarship and essential contributions to building and sustaining the ISIPTA community. Although the basic idea behind imprecise probability is (at least) 150 years old, a mature mathematical theory has only taken full form in the last 30 years. Interest in imprecise probability during this period has also grown, but many of the ideas that the mature theory serves can be difficult to apprehend to those new to the subject. Although these fundamental ideas are common knowledge in the ISIPTA community, they are expressed, when they are expressed at all, obliquely, over the course of years with students and colleagues.
A single essay cannot convey the store of common knowledge from any research community, let alone the ISIPTA community. But, this essay nevertheless is an attempt to guide those familiar with the basic Bayesian framework to appreciate some of the elegant and powerful ideas that underpin the contemporary theory of lower previsions, which is the theory that most people associate with the term ‘imprecise probabilities’
Moving Beyond Sets of Probabilities
The theory of lower previsions is designed around the principles of coherence and sure-loss avoidance, thus steers clear of all the updating anomalies highlighted in Gong and Meng's "Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradox" except dilation. In fact, the traditional problem with the theory of imprecise probability is that coherent inference is too complicated rather than unsettling. Progress has been made simplifying coherent inference by demoting sets of probabilities from fundamental building blocks to secondary representations that are derived or discarded as needed
Independent Natural Extension for Choice Functions
We investigate epistemic independence for choice functions in a multivariate setting. This work is a continuation of earlier work of one of the authors [23], and our results build on the characterization of choice functions in terms of sets of binary preferences recently established by De Bock and De Cooman [7]. We obtain the independent natural extension in this framework. Given the generality of choice functions, our expression for the independent natural extension is the most general one we are aware of, and we show how it implies the independent natural extension for sets of desirable gambles, and therefore also for less informative imprecise-probabilistic models. Once this is in place, we compare this concept of epistemic independence to another independence concept for choice functions proposed by Seidenfeld [22], which De Bock and De Cooman [1] have called S-independence. We show that neither is more general than the other