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Category: new publication

New journal paper: Accept & reject statement-based uncertainty models

Accept & reject statement-based uncertainty models (preprint pdf)
by Erik Quaeghebeur, Gert de Cooman and Filip Hermans

A paper with a long history, polished to perfection. It was conceived out of our frustration with the term ‘desirability’, and its lack of an operational definition. Whereas it is easy to see what it means to accept a gamble, it seems harder to define what it means to find it desirable, or strictly preferable to the status quo. We solve this problem by also looking at what it means to reject a gamble. And in doing so, we come up with a decision-based uncertainty theory that encompasses most of the existing ones in the literature, and which allows us to compare them and put them into proper perspective.

Abstract: We develop a framework for modelling and reasoning with uncertainty based on accept and reject statements about gambles. It generalises the frameworks found in the literature based on statements of acceptability, desirability, or favourability and clarifies their relative position. Next to the statement-based formulation, we also provide a translation in terms of preference relations, discuss—as a bridge to existing frameworks—a number of simplified variants, and show the relationship with prevision-based uncertainty models. We furthermore provide an application to modelling symmetry judgements.

Published where? International Journal of Approximate Reasoning 59 (2015) 61–102.

New journal paper: Coherent predictive inference under exchangeability with imprecise probabilities

Coherent predictive inference under exchangeability with imprecise probabilities (preprint pdf)
by Gert de Cooman, Jasper De Bock and Márcio Alves Diniz

This is what we jokingly refer to as ‘our new book’. It has made me redefine the main goal of my life: to write and publish, if only just once, a paper that is shorter than ten pages.

Abstract: 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 insensitivity—a strengthened version of Walley’s representation invariance—and 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.

Published where? Journal of Artificial Intelligence Research 52 (2015) 1–95.

New edited book: Introduction to Imprecise Probabilities

And the goodies keep coming in: this morning I received advance copies for another Wiley book, Introduction to Imprecise Probabilities, edited by Thomas Augustin, Frank Coolen, Matthias Troffaes and me, but this one took us only about five years to finish.

Introduction to Imprecise Probabilities

To quote from the book description:


In recent years, the theory [of Imprecise Probabilities] has become widely accepted and has been further developed, but a detailed introduction is needed in order to make the material available and accessible to a wide audience. This will be the first book providing such an introduction, covering core theory and recent developments which can be applied to many application areas. All authors of individual chapters are leading researchers on the specific topics, assuring high quality and up-to-date contents.

An Introduction to Imprecise Probabilities provides a comprehensive introduction to imprecise probabilities, including theory and applications reflecting the current state if the art. Each chapter is written by experts on the respective topics, including: Sets of desirable gambles; Coherent lower (conditional) previsions; Special cases and links to literature; Decision making; Graphical models; Classification; Reliability and risk assessment; Statistical inference; Structural judgments; Aspects of implementation (including elicitation and computation); Models in finance; Game-theoretic probability; Stochastic processes (including Markov chains); Engineering applications.

Essential reading for researchers in academia, research institutes and other organizations, as well as practitioners engaged in areas such as risk analysis and engineering.

New book: Lower previsions

This morning I received advance copies from Wiley of the book Lower Previsions that Matthias Troffaes and I have been working on for the past eight years. I am really excited that it is finally here, and I hope that our effort will help other people extend and apply the theory of imprecise probabilities.

Lower Previsions

I hope Wiley won’t mind if I quote here from our own Preface, I certainly don’t:

Lower Previsions is an overview of, and reference guide to, the mathematics of lower previsions. Starting from first principles—acceptability—, we derive their mathematical properties and relate them to a wide range of other uncertainty models—belief functions, Choquet capacities, possibility measures—and mathematical concepts—including filters, limits, propositional logic, integration and many other constructs from functional and convex analysis. The material of the book is advanced and aimed at researchers, postgraduate students and lecturers. It will be of interest to statisticians, probabilists, mathematicians and anyone whose field of interest includes some form of uncertainty modelling, both from a practical and a theoretical point of view.

Work on this book started about 8 years ago. The idea was, at that time, to turn the most important results in Matthias’s PhD thesis, supervised by Gert, into a coherent and more or less self-contained research monograph. Our initial plan was to focus on two things: first, the relationship between natural extension and integration and second, the discussion of lower previsions defined on unbounded gambles. It soon became clear that, in order to make the book more self-contained, we needed to include much more material on lower previsions themselves. At the same time, we gathered from conversations with close colleagues that there was a definite interest in—given the perceived lack of—a comprehensive treatment of the existing theory of lower previsions. And so we decided to include, besides our own, a number of contributions from other people, amongst whom in particular are Peter Williams, Peter Walley, Sebastian Maass, Dieter Denneberg and Enrique Miranda. The present book, therefore, differs significantly from the one we started out with. While initially, the book was mostly focused on Matthias’s PhD work, in its present form, it contains much more material and both authors have contributed to it on an equal footing.

In the first part of this book, we expose and expand on the main ideas behind the theory that deals exclusively with bounded gambles. We also discuss a wide variety of special cases that may be of interest when implementing these ideas in practical problems. In doing so, we demonstrate the unifying power behind the concept of coherent lower previsions, for uncertainty modelling as well as for functional analysis. In the second part of this book, we extend the scope of the theory of lower previsions by allowing it to deal with real gambles that are not necessarily bounded. In that part, we also deal with conditioning and provide practical constructions for extending lower previsions to unbounded gambles.

[…]

We have tried to make this book as self-contained as possible.  This means, amongst other things, that we have tried to at least provide an explicit formulation—if not an actual proof—of most results that we use. We have relegated to a number of appendices supporting material that did not fit nicely into the main storyline.

If you are used to a measure-theoretic approach to probability, you may initially feel somewhat lost in this book, because we do not start out with measurability at all. Indeed, the foundations of lower previsions, for arbitrary spaces, do not rely on any notion of measurability. This may come as a surprise to some people who think that using measurability is natural and should come first. Instead, our discussion of lower previsions is founded on a notion of acceptability of gambles, which has a direct behavioural interpretation. In other words, rather than posing laws of probability, we pose laws of acceptability, from which laws of probability are derived.

As often as possible, we give detailed accounts of most steps in the proofs, with explicit references to other results that are being used. This may appear to be pedantic—or even worse, condescending—to some, but we thought it better to be too specific rather than incur the risk of explaining too little: as this is not a small book, we cannot expect any reader to remember every little result we have proved or mentioned earlier.

[…]

We hope that you will enjoy reading and working with this book as much as we have enjoyed researching and writing it.

New arXiv note: Continuity of imprecise Markov chains with respect to the pointwise convergence of monotone sequences of gambles

Continuity of imprecise Markov chains with respect to the pointwise convergence of monotone sequences of gambles (arXiv page)

by Jasper De Bock and Gert de Cooman

Abstract: The aim of these notes is to study the conditions under which the natural extension of an imprecise Markov chain is continuous with respect to the pointwise convergence of monotone (either non-decreasing or non-increasing) sequences of gambles that are n-measurable, with n a natural number. The framework in which we do this is that of the theory of imprecise random processes currently being developed in (De Cooman, 2014). We find that for non-decreasing sequences, continuity is always guaranteed if the state space of the Markov chain is finite. We find a similar result for non-increasing sequences, under the extra condition that the joint model is constructed using the Ville-Vovk-Shafer natural extension rather than the Williams natural extension. We also discuss the extent to which these results apply to general random processes.

More information?
This is a note containing the proofs of two useful continuity results for monotone adapted processes in imprecise Markov chains. We intend to include them, or rely on them, in forthcoming papers on Markov processes, and on stochastic processes using imprecise probabilities. I will post more bibliographic details on my UGent website as they come in.

New arXiv paper preprint: Accept & Reject Statement-Based Uncertainty Models

Accept & Reject Statement-Based Uncertainty Models (arXiv page)

by Erik Quaeghebeur, Gert de Cooman and Filip Hermans

Abstract: We develop a framework for modelling and reasoning with uncertainty based on accept and reject statements about gambles. It generalises the frameworks found in the literature based on statements of acceptability, desirability, or favourability and clarifies their relative position. Next to the statement-based formulation, we also provide a translation in terms of preference relations, discuss—as a bridge to existing frameworks—a number of simplified variants, and show the relationship with prevision-based uncertainty models. We furthermore provide an application to modelling symmetry judgements.

More information?
I will post more bibliographic details on my UGent website as they come in.

New arXiv paper preprint: An efficient algorithm for estimating state sequences in imprecise hidden Markov models

An efficient algorithm for estimating state sequences in imprecise hidden Markov models (arXiv page)

by Jasper De Bock and Gert de Cooman

Abstract: We present an efficient exact algorithm for estimating state sequences from outputs (or observations) in imprecise hidden Markov models (iHMM), where both the uncertainty linking one state to the next, and that linking a state to its output, are represented using coherent lower previsions. The notion of independence we associate with the credal network representing the iHMM is that of epistemic irrelevance. We consider as best estimates for state sequences the (Walley–Sen) 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 HMMs with precise transition and output probabilities (pHMMs). We argue that the computational complexity is at worst quadratic in the length of the Markov chain, cubic in the number of states, and essentially linear in the number of maximal state sequences. For binary iHMMs, we investigate experimentally how the number of maximal state sequences depends on the model parameters. We also present a simple toy application in optical character recognition, demonstrating that our algorithm can be used to robustify the inferences made by precise probability models.

More information?
I will post more bibliographic details on my UGent website as they come in.

An Operational Approach to Graphical Uncertainty Modelling: PhD Thesis by Filip Hermans

My student Filip Hermans defended his PhD thesis in May of this year. Thus far, I haven’t had time to blog or brag about it, but it is time I made up for that. I believe that what he has done is really worth taking a good look at, especially if you are interested in imprecise probabilities, the foundations of probabilistic inference, or stochastic processes.

There are, besides the Introduction and Conclusion, four main chapters in the thesis. The first deals with acceptability of gambles, and constitutes a valiant attempt at providing a foundation for (imprecise-)probabilistic inference based on weak and strict preference relations. It is the basis for a more recent and more detailed analysis that Erik Quaeghebeur, Filip and I have been working during over the last year (and which I will have occasion to talk about later). All other chapters are built on this framework. The second deals with (imprecise-)probabilistic inference associated with event trees, and provides the foundations for a theory of (discrete-time) stochastic processes using imprecise probabilities. In the third chapter, this is applied in particular to Markov processes. The fourth chapter extends the arguments of the previous two even further to allow for inference in credal networks with a tree structure.

And, following the tradition started by Erik Quaeghebeur in his PhD thesis, there is, of course, a blonde footnote dedicated to Enrique Miranda:

Afbeelding

New conference paper: Conglomerable natural extension

Conglomerable natural extension (preprint pdf)
by Enrique Miranda, Marco Zaffalon and Gert de Cooman

Abstract: We study the weakest conglomerable model that is implied by desirability or probability assessments: the conglomerable natural extension. We show that taking the natural extension of the assessments while imposing conglomerability—the procedure adopted in Walley’s theory—does not yield, in general, the conglomerable natural extension (but it does so in the case of the marginal extension). Iterating this process produces a sequence of models that approach the conglomerable natural extension, although it is not known, at this point, whether it is attained in the limit. We give sufficient conditions for this to happen in some special cases, and study the differences between working with coherent sets of desirable gambles and coherent lower previsions. Our results indicate that it might be necessary to re-think the foundations of Walley’s theory of coherent conditional lower previsions for infinite partitions of conditioning events.

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New conference paper: Independent natural extension for sets of desirable gambles

Independent natural extension for sets of desirable gambles (preprint pdf)
by Gert de Cooman and Enrique Miranda

Abstract: We investigate how to combine a number of marginal coherent sets of desirable gambles into a joint set using the properties of epistemic irrelevance and independence. We provide formulas for the smallest such joint, called their independent natural extension, and study its main properties. The independent natural extension of maximal sets of gambles allows us to define the strong product of sets of desirable gambles. Finally, we explore an easy way to generalise these results to also apply for the conditional versions of epistemic irrelevance and independence.

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New conference paper: State sequence prediction in imprecise hidden Markov models

State sequence prediction in imprecise hidden Markov models (preprint pdf)
by Jasper De Bock and Gert de Cooman

Abstract: We present an efficient exact algorithm for estimating state sequences from outputs (or observations) in imprecise hidden Markov models (iHMM), where both the uncertainty linking one state to the next, and that linking a state to its output, are represented using coherent lower previsions. The notion of independence we associate with the credal network representing the iHMM is that of epistemic irrelevance. We consider as best estimates for state sequences the (Walley–Sen) 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 HMMs with precise transition and output probabilities (pHMMs). We argue that the computational complexity is at worst quadratic in the length of the Markov chain, cubic in the number of states, and essentially linear in the number of maximal state sequences. For binary iHMMs, we investigate experimentally how the number of maximal state sequences depends on the model parameters.

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Master’s theses by Jasper De Bock and Arthur Van Camp

I have been telling you about work Jasper De Bock and I have done on state sequence prediction in imprecise hidden Markov Models, leading to the development of the EstiHMM algorithm. Now, Jasper’s master’s thesis (written in Dutch with an English extended abstract) on this subject has been submitted, and is available for download. We have submitted a paper about this to the ISIPTA 2011 conference.

Arthur Van Camp has been working on applying the MePiCTIr algorithm to inference in imprecise Hidden Markov models, with a simple but interesting application in earthquake rate prediction. Hidden in his text is an interesting idea about the interplay between quantisation (or discretisation) and imprecision I have been toying with for some time now, and hope to be able to work on with him in the coming year. Arthur has submitted an abstract for poster presentation at ISIPTA 2011. His master’s thesis (written in Dutch with an English extended abstract) on this subject has been submitted, and is available for download too.

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New journal paper: Independent natural extension

Independent natural extension (preprint pdf)
by Gert de Cooman, Enrique Miranda and Marco Zaffalon

Abstract: There is no unique extension of the standard notion of probabilistic independence to the case where probabilities are indeterminate or imprecisely specified. Epistemic independence is an extension that formalises the intuitive idea of mutual irrelevance between different sources of information. This gives epistemic independence very wide scope as well as appeal: this interpretation of independence is often taken as natural also in precise-probabilistic contexts. Nevertheless, epistemic independence has received little attention so far. This paper develops the foundations of this notion for variables assuming values in finite spaces. We define (epistemically) independent products of marginals (or possibly conditionals) and show that there always is a unique least-committal such independent product, which we call the independent natural extension. We supply an explicit formula for it, and study some of its properties, such as associativity, marginalisation and external additivity, which are basic tools to work with the independent natural extension. Additionally, we consider a number of ways in which the standard factorisation formula for independence can be generalised to an imprecise-probabilistic context. We show, under some mild conditions, that when the focus is on least-committal models, using the independent natural extension is equivalent to imposing a so-called strong factorisation property. This is an important outcome for applications as it gives a simple tool to make sure that inferences are consistent with epistemic independence judgements. We discuss the potential of our results for applications in Artificial Intelligence by recalling recent work by some of us, where the independent natural extension was applied to graphical models. It has allowed, for the first time, the development of an exact linear-time algorithm for the imprecise probability updating of credal trees.

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New journal paper: Exchangeability and sets of desirable gambles

Exchangeablity and sets of desirable gambles (preprint pdf)
by Gert de Cooman and Erik Quaeghebeur

Abstract: Sets of desirable gambles constitute a quite general type of uncertainty model with an interesting geometrical interpretation. We give a general discussion of such models and their rationality criteria. We study exchangeability assessments for them, and prove counterparts of de Finetti’s finite and infinite representation theorems. We show that the finite representation in terms of count vectors has a very nice geometrical interpretation, and that the representation in terms of frequency vectors is tied up with multivariate Bernstein (basis) polynomials. We also lay bare the relationships between the representations of updated exchangeable models, and discuss conservative inference (natural extension) under exchangeability and the extension of exchangeable sequences.

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New journal paper: Epistemic irrelevance in credal nets: the case of imprecise Markov trees

Epistemic irrelevance in credal nets: The case of imprecise Markov trees (doi:10.1016/j.ijar.2010.08.011, preprint pdf)
by Gert de Cooman, Filip Hermans, Marco Zaffalon and Alessandro Antonucci

Abstract: We focus on credal nets, which are graphical models that generalise Bayesian nets to imprecise probability. We replace the notion of strong independence commonly used in credal nets with the weaker notion of epistemic irrelevance, which is arguably more suited for a behavioural theory of probability. Focusing on directed trees, we show how to combine the given local uncertainty models in the nodes of the graph into a global model, and we use this to construct and justify an exact message-passing algorithm that computes updated beliefs for a variable in the tree. The algorithm, which is linear in the number of nodes, is formulated entirely in terms of coherent lower previsions, and is shown to satisfy a number of rationality requirements. We supply examples of the algorithm’s operation, and report an application to on-line character recognition that illustrates the advantages of our approach for prediction. We comment on the perspectives, opened by the availability, for the first time, of a truly efficient algorithm based on epistemic irrelevance.

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New journal paper: Exchangeable lower previsions

Exchangeable lower previsions (pdf)
by Gert de Cooman, Erik Quaeghebeur and Enrique Miranda

Abstract: We extend de Finetti’s [Ann. Inst. H. Poincaré 7 (1937) 1-68] notion of exchangeability to finite and countable sequences of variables, when a subject’s beliefs about them are modelled using coherent lower previsions rather than (linear) previsions. We derive representation theorems in both the finite and countable cases, in terms of sampling without and with replacement, respectively.

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