100 research outputs found
Irrelevant natural extension for choice functions
We consider coherent choice functions under the recent axiomatisation proposed by De Bock and De Cooman that guarantees a representation in terms of binary preferences, and we discuss how to define conditioning in this framework. In a multivariate context, we propose a notion of marginalisation, and its inverse operation called weak (cylindrical) extension. We combine this with our definition of conditioning to define a notion of irrelevance, and we obtain the irrelevant natural extension in this framework: the least informative choice function that satisfies a given irrelevance assessment
Exchangeable choice functions
We investigate how to model exchangeability with choice functions.
Exchangeability is a structural assessment on a sequence of uncertain
variables. We show how such assessments are a special indifference assessment,
and how that leads to a counterpart of de Finetti's Representation Theorem,
both in a finite and a countable context
Modelling practical certainty and its link with classical propositional logic
We model practical certainty in the language of accept & reject statement-based uncertainty models. We present three different ways, each time using a different nature of assessment: we study coherent models following from (i) favourability assessments, (ii) acceptability assessments, and (iii) indifference assessments. We argue that a statement of favourability, when used with an appropriate background model, essentially boils down to stating a belief of practical certainty using acceptability assessments. We show that the corresponding models do not form an intersection structure, in contradistinction with the coherent models following from an indifferenc assessment. We construct embeddings of classical propositional logic into each of our models for practical certainty
Learning imprecise hidden Markov models
We present a method for learning imprecise local uncertainty models in stationary hidden Markov models. If there is enough data to justify precise local uncertainty models, then existing learning algorithms, such as the Baum–Welch algorithm, can be used. When there is not enough evidence to justify precise models, the method we suggest here has a number of interesting features
Lexicographic choice functions
We investigate a generalisation of the coherent choice functions considered
by Seidenfeld et al. (2010), by sticking to the convexity axiom but imposing no
Archimedeanity condition. We define our choice functions on vector spaces of
options, which allows us to incorporate as special cases both Seidenfeld et
al.'s (2010) choice functions on horse lotteries and sets of desirable gambles
(Quaeghebeur, 2014), and to investigate their connections. We show that choice
functions based on sets of desirable options (gambles) satisfy Seidenfeld's
convexity axiom only for very particular types of sets of desirable options,
which are in a one-to-one relationship with the lexicographic probabilities. We
call them lexicographic choice functions. Finally, we prove that these choice
functions can be used to determine the most conservative convex choice function
associated with a given binary relation.Comment: 27 page
Recent advances in imprecise-probabilistic graphical models
We summarise and provide pointers to recent advances in inference and identification for specific types of probabilistic graphical models using imprecise probabilities. Robust inferences can be made in so-called credal networks when the local models attached to their nodes are imprecisely specified as conditional lower previsions, by using exact algorithms whose complexity is comparable to that for the precise-probabilistic counterparts
Choice functions as a tool to model uncertainty
Our aim is to develop a tool for modelling different types of assessments about the uncertain value of some random variable. One well-know and widely used way to model uncertainty is using probability mass functions. However, such probability mass functions are not general enough to model, for instance, a total lack of knowledge. A very successful tool for modelling more general types of assessments is coherent sets of desirable gambles. These have many applications in credal networks, predictive inference, conservative reasoning, and so on. However, they are not capable of modelling beliefs corresponding to 'or' statements, for example the belief that a coin has two equal sides of unknown type: either twice heads or twice tails. Such more general assessments can be modelled with coherent choice functions.
The first thing we do is relate coherent choice functions to coherent sets of desirable gambles, which yields an expression for the most conservative coherent choice function compatible with a coherent set of desirable gambles. Next, we study the order-theoretic properties of coherent choice functions. In order for our theory of choice functions to be successful, we need a good conditioning rule. We propose a very intuitive one, and show that it coincides with the usual one for coherent sets of desirable gambles, and therefore also leads to Bayes’s rule. To conclude, we show how to elegantly deal with assessments of indifference
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