We can learn (more) about the state a quantum system is in through measurements. We look at how to describe the uncertainty about a quantum system's state conditional on executing such measurements. We show that by exploiting the interplay between desirability, coherence and indifference, a general rule for conditioning can be derived. We then apply this rule to conditioning on measurement outcomes, and show how it generalises to conditioning on a set of measurement outcomes.

Desirability can be understood as an extension of Anscombe and Aumann's Bayesian decision theory to sets of expected utilities. At the core of desirability lies an assumption of linearity of the scale in which rewards are measured. It is a traditional assumption used to derive the expected utility model, which clashes with a general representation of rational decision making, though. Allais has, in particular, pointed this out in 1953 with his famous paradox. We note that the utility scale plays the role of a closure operator when we regard desirability as a logical theory. This observation enables us to extend desirability to the nonlinear case by letting the utility scale be represented via a general closure operator. The new theory directly expresses rewards in actual nonlinear currency (money), much in Savage's spirit, while arguably weakening the founding assumptions to a minimum. We characterise the main properties of the new theory both from the perspective of sets of gambles and of their lower and upper prices (previsions). We show how Allais paradox finds a solution in the new theory, and discuss the role of sets of probabilities in the theory.

In a recent paper Miranda & Zaffalon (2020) some results about compatibility or consistency of coherent sets of gambles or lower previsisons have been derived and it was remarked that these results were in fact results of the theory of information or valuation algebras (Kohlas, 2003). This point of view, however, was not worked out by Miranda & Zaffalon (2020). In this paper this issue is taken up and it is shown that coherent sets of gambles, strictly desirable sets of gambles, coherent lower and upper previsions indeed form idempotent information algebras. Like in group theory, certain results concerning particular groups follow from general group theory, so many known results about desirable gambles, lower and linear previsions are indeed properties of an information algebra and follow from the corresponding general theory. Some of these results are discussed in this paper, but there are doubtless many other properties which can be derived from the theory of information algebra.

Teddy Seidenfeld has been arguing for quite a long time that binary preference models are not powerful enough to deal with a number of crucial aspects of imprecision and indeterminacy in uncertain inference and decision making. It is at his insistence that we initiated our study of so-called sets of desirable option sets, which we have argued elsewhere provides an elegant and powerful approach to dealing with general, binary as well as non-binary, decision-making under uncertainty. We use this approach here to explore an interesting notion of irrelevance (and independence), first suggested by Seidenfeld in an example intended as a criticism of a number of specific decision methodologies based on (convex) binary preferences. We show that the consequences of making such an irrelevance or independence assessment are very strong, and might be used to argue for the use of so-called mixing choice functions, and E-admissibility as the resulting decision scheme.

The latest Industrial revolution has helped industries in achieving very high rates of productivity and efficiency. It has introduced data aggregation and cyber-physical systems to optimize planning and scheduling. Although, uncertainty in the environment and the imprecise nature of human operators are not accurately considered for into the decision making process. This leads to delays in consignments and imprecise budget estimations. This widespread practice in the industrial models is flawed and requires rectification. Various other articles have approached to solve this problem through stochastic or fuzzy set model methods. This paper presents a comprehensive method to logically and realistically quantify the non-deterministic uncertainty through probabilistic uncertainty modelling. This method is applicable on virtually all Industrial data sets, as the model is self adjusting and uses epsilon-contamination to cater to limited or incomplete data sets. The results are numerically validated through an Industrial data set in Flanders, Belgium. The data driven results achieved through this robust scheduling method illustrate the improvement in performance.

Decision making under uncertainty is typically carried out by combining an uncertainty model with a decision rule. If uncertainty is modelled by a probability measure, the by far most popular such decision rule is maximising expected utility, where one chooses the option--or makes the decision--whose expected utility with respect to this probability measure is the highest. Uncertainty can also be modelled in various other ways though. The theory of imprecise probabilities, for example, offers a wide range of extensions of probability theory that provide more flexible modelling possibilities, such as differentiating between stochastic uncertainty and model uncertainty. The most straightforward such extension is to consider a set of probability measures instead of a single one, but one can also use interval probabilities, coherent lower previsions, sets of desirable gambles, belief functions, to name only a few. For all these different types of uncertainty models, various decision rules have been developed, making the total number of possible combinations rather daunting. Choosing which combination of uncertainty model and decision rule to use is therefore difficult and often dealt with in a pragmatic fashion, by using a combination that one is familiar with, that is convenient or that is computationaly advantageous.

We establish the equivalence of two very general theories: the first is the decision-theoretic formalisation of incomplete preferences based on the mixture independence axiom; the second is the theory of coherent sets of desirable gambles (bounded variables) developed in the context of imprecise probability and extended here to vector-valued gambles. Such an equivalence allows us to analyse the theory of incomplete preferences from the point of view of desirability. Among other things, this leads us to uncover an unexpected and clarifying relation: that the notion of `state independence'---the traditional assumption that we can have separate models for beliefs (probabilities) and values (utilities)---coincides with that of `strong independence' in imprecise probability; this connection leads us also to propose much weaker, and arguably more realistic, notions of state independence. Then we simplify the treatment of complete beliefs and values by putting them on a more equal footing. We study the role of the Archimedean condition---which allows us to actually talk of expected utility---, identify some weaknesses and propose alternatives that solve these. More generally speaking, we show that desirability is a valuable alternative foundation to preferences for decision theory that streamlines and unifies a number of concepts while preserving great generality. In addition, the mentioned equivalence shows for the first time how to extend the theory of desirability to imprecise non-linear utility, thus enabling us to formulate one of the most powerful self-consistent theories of reasoning and decision-making available today.

We establish an equivalence between two seemingly different theories: one is the traditional axiomatisation of incomplete preferences on horse lotteries based on the mixture independence axiom; the other is the theory of desirable gambles (bounded random variables) developed in the context of imprecise probability, which we extend here to make it deal with vector-valued gambles. The equivalence allows us to revisit incomplete preferences from the viewpoint, and with the tools, of desirability and through the derived notion of coherent lower previsions (i.e., lower expectation functionals). On this basis, we obtain new results and insights: in particular, we show that the theory of incomplete preferences can be developed assuming only the existence of a worst act--no best act is needed--, and that a weakened Archimedean axiom suffices too; this axiom allows us also to address some controversy about the regularity assumption (that probabilities should be positive--they need not), which enables us also to deal with uncountable possibility spaces; we show that it is always possible to extend in a minimal way a preference relation to one with a worst act, and yet the resulting relation is never Archimedean, except in a trivial case; we show that the traditional notion of state independence coincides with the notion called strong independence in imprecise probability (stochastic independence in the case of complete preferences)--this leads us to give much a weaker definition of state independence than the traditional one; we rework and uniform the notions of complete preferences, beliefs, values; we argue that Archimedeanity does not capture all the problems that can be modelled with sets of expected utilities and we provide a new notion that does precisely that. Perhaps most importantly, we argue throughout that desirability is a powerful and natural setting to model, and work with, incomplete preferences, even in the case of non-Archimedean problems. This leads us to suggest that desirability, rather than preference, should be the primitive notion at the basis of decision-theoretic axiomatisations. Keywords: Incomplete preferences, decision theory, expected utility, desirability, convex cones, imprecise probability.

Coherent reasoning under uncertainty can be represented in a very general manner by coherent sets of desirable gambles. In a context that does not allow for indecision, this leads to an approach that is mathematically equivalent to working with coherent conditional probabilities. If we do allow for indecision, this leads to a more general foundation for coherent (imprecise-)probabilistic inference. In this framework, and for a given finite category set, coherent predictive inference under exchangeability can be represented using Bernstein coherent cones of multivariate polynomials on the simplex generated by this category set. This is a powerful generalisation of de Finetti's Representation Theorem allowing for both imprecision and indecision. We define an inference system as a map that associates a Bernstein coherent cone of polynomials with every finite category set. Many inference principles encountered in the literature can then be interpreted, and represented mathematically, as restrictions on such maps. We discuss, as particular examples, two important inference principles: representation insensitivitya strengthened version of Walley's representation invarianceand specificity. We show that there is an infinity of inference systems that satisfy these two principles, amongst which we discuss in particular the skeptically cautious inference system, the inference systems corresponding to (a modified version of) Walley and Bernard's Imprecise Dirichlet Multinomial Models (IDMM), the skeptical IDMM inference systems, and the Haldane inference system. We also prove that the latter produces the same posterior inferences as would be obtained using Haldane's improper prior, implying that there is an infinity of proper priors that produce the same coherent posterior inferences as Haldane's improper one. Finally, we impose an additional inference principle that allows us to characterise uniquely the immediate predictions for the IDMM inference systems.

We present an efficient exact algorithm for estimating state sequences from outputs or observations in imprecise hidden Markov models (iHMMs). The uncertainty linking one state to the next, and that linking a state to its output, is represented by a set of probability mass functions instead of a single such mass function. We consider as best estimates for state sequences the maximal sequences for the posterior joint state model conditioned on the observed output sequence, associated with a gain function that is the indicator of the state sequence. This corresponds to and generalises finding the state sequence with the highest posterior probability in (precise-probabilistic) HMMs, thereby making our algorithm a generalisation of the one by Viterbi. We argue that the computational complexity of our algorithm is at worst quadratic in the length of the iHMM, cubic in the number of states, and essentially linear in the number of maximal state sequences. An important feature of our imprecise approach is that there may be more than one maximal sequence, typically in those instances where its precise-probabilistic counterpart is sensitive to the choice of prior. For binary iHMMs, we investigate experimentally how the number of maximal state sequences depends on the model parameters. We also present an application in optical character recognition, demonstrating that our algorithm can be usefully applied to robustify the inferences made by its precise-probabilistic counterpart.