March 9th, 2016 by ()

Jasper de Bock’s PhD thesis on Credal Networks under Epistemic Irrelevance

On 13 May 2015, after four years of intensive research under the enthousiastic supervision of Gert de Cooman, I succesfully defended my PhD Thesis, entitled “Credal Networks under Epistemic Irrelevance: Theory and Algorithms”. The jury was composed of Fabio Cozman, Enrique Miranda, Serafín Moral, Joris Walraevens, Dirk Aeyels, Dries Benoit, Jan Van Campenhout and Rik Van de Walle.

Very brief summary

My dissertation presents a detailed study of credal networks under epistemic irrelevance [1], which are probabilistic graphical models that can compactly and intuitively represent the uncertainty that is associated with the key variables in some domain, and which can then be used to answer various domain-specific queries (compute inferences) that are of interest to the user. They share many of the nice features of Pearl’s celebrated Bayesian networks [2], but have the added advantage that they can represent uncertainty in a more flexible and realistic way.

Unlike credal networks under strong independence, which generalise Bayesian networks in a similar way, very little was so far known about credal networks under epistemic irrelevance and their properties, and efficient inference was possible only for specific inferences in networks with a tree structure [3]. The goal of my PhD was to change this, that is, to study the properties of credal networks under epistemic irrelevance, and to use these properties to develop efficient inference algorithms for them. If you don’t feel like reading through the more detailed summary that I am about to provide, let me just say that I did indeed achieve this goal: I have unraveled the theoretical properties of credal networks under epistemic irrelevance and I have developed multiple efficient exact inference algorithms for them.

Modelling uncertainty

Since a credal network under epistemic irrelevance is a special type of imprecise probability model, my dissertation starts with a general introduction to the theory of imprecise probabilities [4]. Simply put, whenever it is infeasible to reliably estimate a single probability, this theory allows for the use of a set of probabilities instead, each of whose elements is regarded as a candidate for some ideal ‘true’ probability. However, this simplified view is only one of the many ways to look at or interpret imprecise probabilities. In fact, uncertainty can be expressed without any reference to probabilities, using other imprecise-probabilistic frameworks such as sets of desirable gambles, lower previsions and sets of linear previsions. In the first part of my dissertation, I provide a detailed overview of these different frameworks and their interpretation, and discuss how they are connected to each other. I pay special attention to conditional models, which I regard as primitive concepts whose connection with unconditional models should be established by means of rationality criteria. The main advantage of the resulting so-called coherent conditional models is that they do not suffer from the traditional problems that arise when some of the conditioning events have probability zero. This is especially important in the context of imprecise probabilities, where probability zero cannot be ignored because it may easily happen that an event has lower probability zero but positive upper probability.

Updating and conditioning

Although my overview of imprecise probability theory contains new results that fill some gaps in the literature, its contribution mainly consists in bringing together results from various existing frameworks and connecting them to each other.The first real contribution of this dissertation is my discussion of updating, which is the act of changing a model based on the information that some event has occurred. In probability theory, it has become standard practice to do this by conditioning on that event using Bayes’s rule. Similarly, in an imprecise-probabilistic setting, updating is typically performed by applying an imprecise conditioning rule such as regular or natural extension. However, little argumentation is usually given as to why such an approach would make sense. I help address this problem by providing a firm philosophical justification for using natural and regular extension as updating rules. What makes this justification especially powerful is that I derive it directly in terms of sets of desirable gambles. In this way, I avoid making some of the unnecessarily strong assumptions that are traditionally adopted, such as the existence of an ideal ‘true’ probability mass function.

Multivariate models

In order to apply imprecise probabilities in a multivariate context, additional tools are needed, such as marginalisation, as well as ways of combining these tools with concepts such as conditioning and updating. All of this is well known and relatively easy in terms of probabilities, but it becomes more challenging for some of the other imprecise-probabilistic frameworks that I consider. My dissertation devotes a complete chapter to these issues. I gather the existing tools, add new ones whenever something is missing, and connect all of them with one another. The result is a complete and well-founded theory of multivariate imprecise probabilities that is, to the best of my knowledge, novel in its completeness, generality and consistency. Using this theory, I then formally introduce one of the most important concepts of my dissertation: epistemic irrelevance, which is an imprecise-probabilistic generalisation of independence that is assymetric. I recall several existing definitions for this notion, argue why only one of them is really adequate, and compare epistemic irrelevance to other imprecise-probabilistic independence notions. Finally, I explain how structural assessments such as epistemic irrelevance can be combined with direct or local partial probability assessments to construct a multivariate uncertainty model.

Credal networks under epistemic irrelevance

The rest of my dissertation is concerned with one particular type of multivariate uncertainty model: the irrelevant natural extension of a credal network under epistemic irrelevance. The basic idea is very similar to that of a Bayesian network. The starting point is a collection of domain-specific variables that are connected by means of arrows that reflect how these variables depend on each other. The arrows form a Directed Acyclic Graph (DAG), which simply means that there are no directed cycles. The interpretation of the graph is that for any variable, conditional on its parents, its non-parent non-descendants are epistemically irrelevant. Each variable also has a given local imprecise-probabilistic model, conditional on the values of its parents in the graph. In combination with the assessments of epistemic irrelevance that correspond to the DAG, these local models form a credal network under epistemic irrelevance. The most conservative global uncertainty model that is compatible with all these assessments is called the irrelevant natural extension of the network. This concept was first introduced by Cozman [1], who defined it in terms of sets of probabilities under the simplifying assumption that all probabilities are strictly positive. I drop this positivity assumption and provide definitions in terms of three other frameworks as well: sets of desirable gambles, lower previsions and sets of linear previsions. These different definitions turn out to be closely related, which allows me to translate results that are proved in one framework to analogous results in other frameworks.

Why I studied this type of credal networks

Credal networks under epistemic irrelevance are not the only imprecise-probabilistic generalisations of Bayesian networks. In fact, they are not even all that popular. Most authors prefer to consider a different type of credal networks, called credal networks under strong independence. I believe that the main reason for this lack of popularity of credal networks under epistemic irrelevance has been a profound lack of known theoretical properties.  Surely, this has severely inhibited the development of tractable inference algorithms. In fact, untill now, only one inference algorithm was available, and even then, only for a particular type of inference and for networks whose DAG has a tree structure [3]. Nevertheless, due to the remarkable efficiency of this particular algorithm, which is linear in the size of the network, and because that same inference problem is NP-hard in credal networks under strong independence [5], credal networks under epistemic irrelevance are regarded as a promising alternative that requires—and deserves—further research [6, Section 10.6]. This further research is what my dissertation is all about.

Theoretical properties

One of the main contributions of my dissertation is a detailed study of the theoretical properties of the multivariate uncertainty model that corresponds to a credal network under epistemic irrelevance: the irrelevant natural extension. By focusing on the framework of sets of desirable gambles, I was able to derive some remarkable properties of this model, which I then managed to translate to other frameworks as well. A first important example is a fundamental separating hyperplane result that establishes a connection between the irrelevant natural extension of a complete network and that of its subnetworks. This result leads to various marginalisation, factorisation and external additivity properties. A second important result is that the irrelevant natural extension satisfies a collection of epistemic irrelevancies that is induced by AD-separation, an asymmetric adaptation of d-separation that is proved to satisfy all graphoid properties except symmetry. I also establish connections with the notions of independent natural extension and marginal extension and study the updated models that are obtained by applying regular extension to the irrelevant natural extension of a credal network.

Inference algorithms

In the final part of my dissertation, I show how the theoretical properties that I have proved can be used to develop efficient inference algorithms for credal networks under epistemic irrelevance. A first important contribution consists of two preprocessing techniques that can be used to simplify inference problems before the actual algorithm is applied. I explain how and when it is possible to translate an inference problem in a large network into a similar problem in a smaller network, and show how solving a conditional inference problem can be reduced to solving a series of unconditional ones. In a second set of results, I rephrase inference as a linear optimisation problem. As was already mentioned by Cozman [1], every unconditional inference can be computed by solving a linear program. However, in order to establish this result, he required a simplifying positivity assumption. I show that this positivity assumption is not needed; unconditional inferences can always be characterised as the solution of a linear program. For conditional inferences, multiple such linear programs need to be solved. Unfortunately, the size of these linear programs is exponential in the size of the network and this in principle generally applicable method is therefore only tractable for small networks. For the specific case of a network that consists of two disconnected binary variables, I was able to solve the corresponding linear program symbolically. In this way, I obtained closed-form expressions for the extreme points of the independent natural extension of two binary models. Fortunately, the intractability of brute force linear programming methods can often be circumvented by developing other, more efficient and often recursive computational techniques. I illustrate this by means of a number of examples. My most important algorithmic contribution, and the proverbial icing on the cake, is a collection of recursive algorithms that can efficiently compute various inferences in credal networks under epistemic irrelevance whose graphical structure is a recursively decomposable DAG, which is a new type of DAG that includes trees as a special case.


The main conclusion of my dissertation is that credal networks under epistemic irrelevance satisfy surprisingly many powerful theoretical properties, and that these properties can be exploited to develop efficient exact inference algorithms, for large classes of inference problems that were previously presumed to be intractable. Since many of these inference problems are NP-hard in credal networks under strong independence, our results turn credal networks under epistemic irrelevance into a serious, practically feasible alternative type of credal network that should enable practitioners to solve real-life problems for which the corresponding necessary inferences were hitherto regarded as intractable.


[1] Fabio G. Cozman. Credal networks. Artificial Intelligence, 120(2):199– 233, 2000.
[2] Judea Pearl. Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann, San Mateo, 1988.
[3] Gert de Cooman, Filip Hermans, Alessandro Antonucci, and Marco Zaffalon. Epistemic irrelevance in credal nets: the case of imprecise Markov trees. International Journal of Approximate Reasoning, 51(9):1029–1052, 2010.
[4] Peter Walley. Statistical reasoning with imprecise probabilities. Chapman and Hall, London, 1991.
[5] Denis D. Mauá, Cassio P. de Campos, Alessio Benavoli, and Alessandro Antonucci. Probabilistic inference in credal networks: new complexity results. Journal of Artificial Intelligence Research, 50:603–637, 2014.
[6] Alessandro Antonucci, Cassio P. de Campos, and Marco Zaffalon. Probabilistic graphical models. In Thomas Augustin, Frank P. A. Coolen, Gert de Cooman, and Matthias C. M. Troffaes, editors, Introduction to Imprecise Probabilities, pages 207–229. John Wiley & Sons, Chichester, 2014.

About the author

Jasper De Bock is currently a Post-Doctoral Researcher of the Research Foundation – Flanders (FWO), working at the Electronics and Information Systems department of Ghent University. On 13 May 2015, he obtained his PhD in Mathematical Engineering at the same university.

November 6th, 2015 by ()

A report on WPMSIIP’2015, by Christoph Jansen and Julia Plass

The 8th Workshop on Principles and Methods of Statistical Inference with Interval Probability, organized by the working group “Foundations of Statistics and Their Applications” of the Department of Statistics and the “Munich Center for Mathematical Philosophy (MCMP)” of the Department of Philosophy from the LMU Munich took place from the 1st to the 6th of September 2015 (actually we ended at the 5th, because unfortunately our excursion was cancelled because of bad weather forecast…).

In the old WPMSIIP tradition, on every day there were contributions ranging from well-structured presentations to spontaneous discussions of questions. Further, we completed every day with a visit in some nice restaurant (including “Prinz Myshkin”,  Frank’s favourite).

On Tuesday it was Philosophers’ day, who discussed the phenomenon of dilation.
First Seamus Bradley from the MCMP motivated the topic and gave an introduction to “Dilation and the value of information”. Afterwards, Gregory Wheeler and Arthur Paul Pedersen gave some deeper insights by interactively presenting some new ideas from the field. After lunch, it was discussion time. Here, the discussion (which was masterfully moderated by Greg Wheeler and Paul Pedersen) ranged from hot topics like “examples for non-convex sets of probabilities beyond coin-flipping” to more general considerations such as the role of imprecise probabilities in statistics.

On Wednesday, we turned to some more statistical stuff: Imprecise observations. Firstly, Julia Plass gave a presentation on “Statistical modelling under epistemic data imprecision” and discussed some open questions with the audience. Subsequently, Paul Fink introduced us into the art of playing darts: In his talk he proposed a way to generalize the
(illustration of the) NPI model to coarse data by using a dartboard instead of a probability wheel. This was followed by a lively discussion. In the afternoon, Georg “Giorgio” Schollmeyer talked about structural properties of the rounding mapping and some implications for location measures.  To complete the formal part of the day, we heard some new ideas about “MLE-equivalence and Coarse Data in Generalized Linear Models” by Thomas Augustin, who intends (as a main goal) to generalize the concepts of sufficiency to an IP-based treatment of coarse data.

On Thursday, it was Open Topics Day. Paolo Vicig started with a very welcome tutorial talk on “2-coherent and 2-convex Lower Previsions”, which included a detailed version of his ISIPTA ’15 contribution. Afterwards, Thomas Augustin linked to his WPMSIIP 2014 contribution in Ghent by presenting proposals for solutions for questions that arised in the earlier talk. After having solved his “dog problem” (those who were there will know exactly what we’re talking about), Paul Pedersen joined the group after lunch with a spontaneous
presentation about “Strictly coherent choice”. In the end of the day, Christoph Jansen considered some connections between  “Gamma-maximin and least favourable prior

Friday was reserved for Graphical Models. The first talk was jointly given by Barbara Osimani and Roland Poellinger who addressed some applications in the field of “Nesting Causal Models and Evidential Relations”. Again the talk was accompanied by exciting discussions. Next, it was Eva Endres’ turn. Her talk was about “Probabilistic Graphical Models for Statistical Matching” with a focus on discussing some interesting new questions of how to combine (imprecise) probabilistic  graphical models with methods of statistical matching. To round the (academic part of the) day off, Jasper de Bock enthusiastically recalled some concepts of independence for graphical models under the light of sensitivity analysis.

The week ended with a section on Risk and Reliability. Gero Walter presented some open questions from his research on system reliability (revisiting and extending his ISIPTA ’15 contribution) causing some really productive and helpful discussion with the audience. After that there was a joint presentation given by Jasper de Bock, Thomas Krak and Stavros Lopatatzidis. They intensively made use of the blackboard for giving insights to the theory of Imprecise continous time Markov Chains  with a discrete state space, which ranged from theoretical aspects to computational issues. The afternoon was organized by Ulrika Sahlin. The topic of interest was Evidence-based Decision Theory and in particular how to reflect low evidence via imprecision in the result.

Our impression on WPMSIIP 2015: The worky-shoppy ratio was close to (or even larger than) one. A lot of nice presentations were given and interesting ideas were exchangend.
Organizing the workshop was real fun and thus we can only suggest this job to everyone!

If you wish to do so, please contact Frank Coolen.
For general information see also:

See you there (whereever that will be…)!

About the authors

Christoph Jansen and Julia Plass are PhD students of Thomas Augustin at the Department of Statistics, LMU Munich. Christophs’s research interests include decision making under complex uncertainty and Julia’s research focus is on coarse data, partial identification and survey statistics.

November 2nd, 2015 by ()

A report on the 9th SIPTA conference on imprecise probabilities, by Serena Doria

The 9th International Symposium on Imprecise Probability: Theories and Applications was held from Monday 20 to Friday 24 July 2015 in the Italian city of Pescara.

This conference was a result of the productive cooperation among the members of the Steering  Committee, formed by Thomas Augustin, Gert de Cooman, Enrique Miranda, Erik Quaeghebeur, Teddy Seidenfeld, and myself.  I wish to thank  the members of the Local Organizing Committee, Attilio Grilli, Mariangela Scorrano and Andrea Di Cencio for their valuable help and the Department of Engineering and Geology of the University G. d’Annunzio for its financial support.

As with previous ISIPTA meetings, there were only plenary sessions in the program. In total, 31 papers were presented by a short talk and a poster, and 17 additional posters were presented to introduce  preliminary ideas and challenging applications for which the research is not yet completed.

The contributions presented  a large number of new results in different  domains: decision making, statistical inference, belief aggregation, artificial intelligence, and stochastic processes, amongst others.

We were pleased to have three eminent invited speakers:

  • Itzhak Gilboa, from Tel Aviv University and HEC Paris, proposed a unified model of inductive reasoning;
  • Peter Williams, from the University of Sussex and BW Mining, reviewed the intellectual background for the development of coherent lower previsions;
  • Massimo Marinacci, from Bocconi University, discussed approaches to model uncertainty in decision problems.

We were also pleased to have two tutorials to highlight specific subdomains of the wide field of imprecise probability:

  • Barbara Vantaggi, from Universitá “La Sapienza” di Roma, gave a  lecture on de Finetti’s work on coherence and its extensions to an imprecise context;
  • Gregory Wheeler, from Ludwig-Maximilians Universität in Munich,  taught us about the philosophical foundations of imprecise probabilities.

During the conference two sets of prizes were awarded: the Best Poster Award, sponsored by Springer and Wiley, and the IJAR Young Researcher Award, granted by the International Journal of Approximate Reasoning.

The winners of the IJAR Young Researcher Award were

Gold Award

  • Jasper De Bock
  • Denis Deratani Mauà

Silver Award

  • Marco de Angelis
  • Hailin Liu

Honorable Mention

  • Georg Schollmeyer

and the Poster Award Laureates were

  1. Gero Walter, Frank P. A. Coolen, Simme Douwe Flapper: System reliability estimation under prior-data conflict
  2. Gero Walter, Christoph Jansen, Thomas Augustin: Updated network analysis of the imprecise probability community based on ISIPTA electronic proceedings
  3. Erik Quaeghebeur, Chris Wesseling, Emma Beauxis-Aussalet, Teresa Piovesan, Tom Sterkenburg: Eliciting sets of acceptable gambles — The CWI World Cup competition (prize declined)
  4. Julia Plass, Thomas Augustin, Marco E. G. V. Cattaneo, Georg Schollmeyer: Statistical modelling under epistemic data imprecision: some results on estimating multinomial distributions and logistic regression for coarse categorical data
  5. Arthur Van Camp, Gert de Cooman, Enrique Miranda, Erik Quaeghebeur: Modelling indifference with choice functions
  6. Julia Plass, Paul Fink, Norbert Schöning, Thomas Augustin: Statistical modelling in surveys without neglecting the undecided: multinomial logistic regression models and imprecise classification trees under ontic data imprecision
  7. Fabio Gagliardi Cozman: Some remarks on sets of lexicographic probabilities and sets of desirable gambles

About the author

Serena Doria is researcher in Probability and Mathematical Statistics at the Department of Engineering and Geology of the University G.d’Annunzio, Chieti-Pescara, Italy. Her research interests are coherent upper conditional previsions, Hausdorff outer measures, Choquet integral, fractal sets.

January 5th, 2015 by ()

A report on the 6th SIPTA school on imprecise probabilities

The sixth SIPTA school on imprecise probabilities was held in the beautiful city of Montpellier (France), on July 21-25, 2014.The school was composed of twenty five participants, from no less than ten different coutnries. It was organized by Kevin Loquin from LIRMM in Montpellier, with the collaboration of Olivier Strauss, Erik Quaeghebeur and Enrique Miranda.

The school was held in the beautiful botanic institute, built in the end of the nineteenth century, situated just next to the oldest french botanic garden.

The school started with the usual introduction to imprecise probability models, given by Sébastien Destercke and Enrique Miranda, during which students learned about lower previsions and about drawing convex sets of probabilities. It was   followed by a course on graphical models and algortihms and approximation methods for imprecise probabilities imparted by Cassio de Campos, Alessandro Antonucci and Francesca Mangili, from the IDSIA research institute in Lugano. This was the occasion to show some possible applications of the monday morning theory.

Tuesday afternoon was time to relax, with a guided tour on the Botanic garden in Montpellier for those who wanted to join. This was the occasion to know all about the history of the garden, as well as of some of its plants. The visit to the glass house under the Montpellier sun was also something to remember. After this cultural event, some of the participants took place in a sports session playing football and volleyball, a good occasion to let off some steam. Finally, the gala dinner at the Oceania Hostel was the occasion to enjoy good food and company in a relaxed atmosphere.

Wednesday morning was the occasion for the students to briefly present their on-going work and projects, in order to grab some pieces of advice from the teachers. After this exchange, Matthias Troffaes from Durham University gave a lecture starting with the basics of decision making with imprecise probabilities and finishing with dynamical decision making.

On Thursday, Gert de Cooman from Ghent University and Erik Quaeghebeur from the CWI in the Netherlands discussed how to make inference with coherent lower previsions, and Inés Couso and Didier Dubois presented other imprecise probability models alternative to coherent lower previsions.

Friday was the good time to take some pictures of the participants, and to finish with less theory and more practice. The morning was devoted to applied topics, such as climate change, environmental risk assessments and tomography, while the afternoon was devoted to a revision session where the students could ask additional questions to the speakers. This session was the occasion to give extra explanation, or to re-explain some notions that were only quickly browsed or ill-understood during the week. It was also quite useful to know what could be improved for the next summer school!
The school was sponsored by LIRMM, the University of Montpellier, Labex Numev and the Pôle de Recherche et d’enseignement supérieur Sud de France. More detailed information about the school as well as the materials from the different sessions can be found at the school website.


November 11th, 2014 by ()

A report on the book ‘Random sets and random fuzzy sets as ill-perceived random variables’



Random sets and random fuzzy sets as ill-perceived random variables. An introduction for PhD students and practitioners, by Inés Couso, Didier Dubois and Luciano Sánchez. Springer Briefs in Applied Sciences and Technology, subseries: Springer Briefs in Computational Intelligence. 2014.


The notion of random set has been around for about 50 years, and has been considered rather independently by two sets of prominent authors. There are the works published in the mid-sixties by well-known economists like Aumann and Debreu on the integration of set-valued functions, followed by a full-fledged mathematical development by Kendall and Matheron. Completely apart from this trend is the pioneering work of Dempster who studied upper and lower probabilities induced by a multivalued mapping from a probability space to the range of an attribute of interest. On this basis, Shafer theory of evidence was built. Formally this is an avatar of random set theory.

However in the first “ontic” view,  random sets are considered as precise representations of random complex entities whose outcomes were naturally described by sets of elements. In contrast, Dempster adopts an epistemic stance, and interprets a random set as representing incomplete information on an ill-known random variable. The theory of belief functions adopts a similar understanding to describe a non-additive extension of subjective probabilities that accommodates, again, incomplete information. Besides, the literature contains many papers about fuzzy random variables since the late seventies, where again authors can be shared between ontic and epistemic schools, but the majority of recent works adopts and extend the “ontic” point of view of random sets.

This short book focuses on the use of random sets and fuzzy random variables as natural representations of ill-observed variables or partially known conditional probabilities.  It deals with the use of random sets and fuzzy random variables from an epistemic point of view.  It is closely connected to the possibilistic interpretation of fuzzy sets, suggested by L. A. Zadeh in 1978. Within this context, the relation between possibility measures and families of nested confidence intervals, and their relation with cuts of fuzzy sets was independently studied by Inés Couso and Luciano Sánchez, and Didier Dubois and colleagues during the last decade. The terms ‘‘ontic’’ and ‘‘epistemic’’ fuzzy sets were suggested by D. Dubois and H. Prade in order to distinguish between complex set-valued entities and the representation of partial knowledge about point-valued entities.

The content of this book is based on three chapters, completed with examples and exercises.

The first chapter proposes a genealogy of published works on random sets, fuzzy sets and belief functions since the mid-sixties, explaining the development and the cross-fertilization of the various trends. The second chapter develops the epistemic view of random sets, and their various representations in terms of families of probability functions. It also questions the relevance of classical approaches to random sets that evaluate scalar statistical parameters like variance. The third chapter does the same job with fuzzy random variables understood either as epistemic fuzzy sets of random variables, or as credal sets induced by  conditional possibility functions.

The aim of the book is to emphasize the difference between the ontic and epistemic views of (fuzzy) random sets, so that students or users working with them may choose the correct approach in a given problem. So its contribution is more philosophical and methodological than a formal exposition of the mathematics underlying the subject-matter. It is motivated by the existence of numerous published papers where it is not clear what random sets mean in practice, and where the motivations (typically epistemic: handling imprecision) often look at odds with the proposed approach (often based on the classical statistical approach to random fuzzy sets as a special kind of functional data).

About the authors

Inés Couso is Associate Professor with the Department of Statistics at the University of Oviedo and member of several journal committees such as International Journal of Approximate Reasoning of Fuzzy Sets and Systems. Her research interests include random sets, fuzzy random variables, possibility theory, independence notions in imprecise probabilities, information fusion and statistics with low quality data.

Didier Dubois is CNRS Research Advisor at the Institut de Recherche en Informatique de Toulouse (IRIT), and co-editor in Chief of Fuzzy Sets and Systems. His research interests include possibility theory, fuzzy sets, qualitative decision theory and information fusion, among many others.

Luciano Sánchez is Full Professor with the Department of Computer Sciences at the University of Oviedo. He is the head of the research group Metrology and Models and a founding partner of the spin-off of the research group, IDALIA S.L. His research interests include include algorithms for intelligent data analysis and their application to practical problems of industrial modeling with low-quality data.

October 27th, 2014 by ()

A report on WPMSIIP’2014, by Arthur van Camp and Stavros Lopatatzidis

The 7th edition of the Workshop on Principles and Methods of Statistical Inference with Interval Probability (abbreviated as WPMSIIP in the most pronounceable way), organised by the SYSTeMS Research group of Ghent University, took place in Ghent from 8 until 12 September 2014.

There were 17 participants from 6 countries.

Every morning and afternoon had its own session in which 2 or 3 persons presented something about a specific topic, in order to stir up the debate, allowing for ample time to discuss and work together.

The first session, on Monday morning, was devoted to statistics. The Institut für Statistik group from the Ludwig Maximilians-Universität Munich led the discussion. All three enthusiastic presenters, Gero Walter, Georg Schollmeyer and Julia Plaß, have their roots in LMU.

In the afternoon, the session was about classification. Unfortunately, prof. Lev Utkin could not make it to Ghent, but luckily Frank Coolen and Paul Fink led the debate lively.

The next session, on Tuesday morning, was about choice functions. Also very unfortunately, Paul Pedersen was unable to come to Ghent. Arthur Van Camp stirred up the discussion about choice functions, and Gert de Cooman filled the gap with a presentation about the foundations of parametric predictive inference.

The next session was foundation, in the afternoon. Enrique Miranda enlightened the audience with a technical discussion about independent products on infinite spaces, Marco Cattaneo warned us about consequences of updating with imprecise probability models, and Erik Quaeghebeur puzzled our minds with a philosophical problem called “Sleeping beauty”.

Wednesday morning was reserved for open topics. Thomas Augustin gave a nice overview of the history of statistics with imprecise probabilities, Qianru Ge introduced an application in reliability optimisation, and Vladimir Vovk talked about the invalidity of NPI.

In the afternoon, we all went to the Ghent city brewery Gruut, where they introduced us both theoretically and pratically in the brewing process. After that, we discovered Ghent from another side: from the side of the water. We sailed with two boats along the inner canals of Ghent.




On Thursday morning there was a session about bivariate models and copulas.

Frank Coolen showed us how to use copulas for inference for bivariata data, and Enrique Miranda “crushed” the podium with results on bivariate p-boxes. Both speakers granted us with lots of interesting discussion material.

After that, in the afternoon, we had a session about credal networks. Cedric De Boom presented the work he did for his master’s thesis, and Jasper De Bock saddled us with many challenging questions regarding independence.

The last day started with a session about stochastic processes, where Francesca Mangili discussed with us about Bayesian nonparametric methods for hypothesis testing. The session ended with Stavros Lopatatzidis who introduced imprecise queueing systems, thereby using the public as a real life example.

The last session of this WPMSIIP was about game-theoretic probability. With Vladimir Vovk presenting material on conformal predictions, and Gert de Cooman introducing a new point wise ergodic theorem, we had two very interesting speakers to end the seventh WPMSIIP.

We have the feeling that the workshop provided food for discussion, revealed several aspects of imprecise probabilities in real life—such as the betting framework— and created the opportunity to better know our international colleagues a bit more informally. We were very pleased to organise this workshop, together with Gert and Jasper.

About the authors

Arthur van Camp and Stavros Lopatatzidis are PhD students at SYSTeMS Group within Ghent University. Research Group at Ghent University. Stavros’ research interests include artificial intelligence, and especially probabilistic models, game theory, queueing theory and multi-agent systems. Arthur’s research interests are in choice functions, and imprecise probability theory in general.

July 30th, 2014 by ()

A report on the Workshop Imprecise Probabilities in Statistics and Philosophy, by Seamus Bradley

A workshop on Imprecise Probabilities in Statistics and Philosophy took place at LMU Munich on the 27th and 28th of June. The workshop was co-organised by the Munich Center for Mathematical Philosophy and the LMU statistics department. There were speakers from four continents, and a broad range of views in philosophy and statistics were represented. The conference was a great success and we hope that this leads to closer ties between the philosophy and statistics communities.

The conference opened with the first keynote talk by Teddy Seidenfeld who discussed two criteria for coherence of personal probabilities and their extensions to Imprecie Probabilities (IP). Next, Carl Wagner discussed an extension of Jeffrey conditioning to more general kinds of evidence. Frank Coolen then discussed non-parametric predictive inference which naturally gives rise to sets of probabilities. Catrin Campbell-Moore showed how IP arises when attempting to give a semantics for self-referential probabilities.

Brian Hill argued that the standard dynamic choice argument against non-expected utility theories is mistaken. Arthur Paul Pedersen and Gregory Wheeler characterised the conditions under which a set of probabilities is subject to dilation. Frederik Herzberg discussed aggregation of infinitely many probability judgements. The first day of the conference closed with Arthur van Camp building bridges between approaches to rational belief based on desirable sets of gambles and choice functions.

The second keynote speaker, Fabio Cozman, opened day two of IPSP.
He discussed the difficulties with finding a concept of independence for IP that satisfies standard graph-theoretical assumptions. Yann Bennetreau-Dupin pointed out that the problem with “noninformative” (precise) priors being too informative can be overcome with IP and thereby solve paradoxes like the Doomsday paradox. Jan-Willem Romeijn discussed how to develop a theory of when statistical information sanctions full belief. Anthony Peressini used interval analysis applied to imprecise chances to avoid some problems with the discontinuous evolution of chance.

Marco Cattaneo used a measure based on likelihoods to give some content to the “reliability index” in Gärdenfors and Sahlin’s Unreliable Probabilities model. Seamus Bradley argued that two prima facie problems for updating IP aren’t problems once the proper interpretation of IP is used. Namjoong Kim discussed another problem for IP updating. The conference closed with our final keynote speaker, James M. Joyce, who discussed using scoring rules to model an agent’s epistemic values (e.g. an agent’s attitude to epistemic risk).

The workshop was supported by the Alexander von Humboldt Foundation, the LMU Statistics department and the LMU Universitätsgesellschaft.
The keynote talks were filmed and the videos are available online through the media page of the conference website.

About the author

Seamus Bradley is a postdoctoral research fellow at the Munich Center for Mathematical Philosophy at LMU Munich. His research interests are in philosophical theories of rational belief and decision, and philosophy of science. More information can be found at

July 17th, 2014 by ()

Two New Books on Imprecise Probability Theories, by Matthias C. M. Troffaes [1]

[1] Thanks to Gert and Frank for proofreading this blog post.









The Books

Recently, two books have been published on imprecise probability theory:

  1. Lower Previsions: a monograph on said subject by myself and Gert de Cooman.
  2. Introduction to Imprecise Probabilities: a collection of contributed chapters on a wide range of topics, edited by Thomas Augustin, Frank Coolen, Gert de Cooman, and myself, with contributions from Joaquín Abellán, Alessandro Antonucci, Cassio P. de Campos, Giorgio Corani, Sébastien Destercke, Didier Dubois, Robert Hable, Filip Hermans, Nathan Huntley, Andrés Masegosa, Enrique Miranda, Serafín Moral, Michael Oberguggenberger, Erik Quaeghebeur, Glenn Shafer, Damjan Skulj, Michael Smithson, Lev Utkin, Gero Walter, Vladimir Vovk, Paolo Vicig, and Marco Zaffalon.

A monumental effort has gone into both publications. Summaries of their content, with tables of contents, can be found on Wiley’s website, linked above (just click on either title). Gert’s blog also replicates part of the preface of the lower previsions book. In this blog post here on the SIPTA site, I thought it would be useful to provide some insight into why these books have come about at all, why their publication is important, reflect back on how we eventually got there, and perhaps on what the next challenges might be.

But before I do that, I would like to express my enormous thanks to everyone who has supported these books, directly or indirectly, foremost my co-author Gert, co-editors Thomas and Frank, and all contributors named above, but also anyone who has helped the books along from the sidelines, our families, and many of our colleagues (in particular Teddy Seidenfeld), whose advice and help through the years have been invaluable.

Lower Previsions

If I want to go back as far as I can, the Lower Previsions book actually started around 2002, about a year after I started my PhD. Of course, initially, we had no clue yet a book was being conceived. It started with Gert giving me a “toy problem”: extend the theory of lower previsions to unbounded random variables. In the process of solving this problem, a report was written documenting various technical results. Unfortunately I no longer have an exact record of these early versions of the report. But in our paper, “Extension of coherent lower previsions to unbounded random variables” submitted to IPMU 2002, reference [3] says: “Gert de Cooman and Matthias Troffaes. Lower previsions for unbounded random variables. Book, in progress.” Apparently we got less ambitious shortly after. In September 2002, in a paper submitted to SPMS 2002, reference [4] says: “Gert de Cooman and Matthias C. M. Troffaes. Lower previsions for unbounded random variables. Tech. report, [...], in progress.” However, the thought of writing a book was certainly there. As it goes, PhDs are full of distractions, which led to neither book nor report ever being published before the end of my PhD in 2005. Most of the material for the report simply ended up in some way as part of my PhD dissertation.

Around the time of my PhD defence, Gert strongly encouraged me to get my PhD out in book form, particularly the results on unbounded random variables. Leaving these results buried inside the PhD would have been a waste indeed. Moreover, we had many good ideas that we still wanted to explore. So, in July 2005, about a month after I arrived in the US for a year of post-doctoral research with Teddy Seidenfeld, I wrote to Gert (in Dutch, here translated to English):

Subject: hello

[...] Anyway, enjoy the vacation! In the mean time I am trying as soon as possible to get the book about unbounded gambles into an acceptable form. [...]

I honestly did not quite think it would take nine more years. :-)

Since 2006, I have been keeping all my files under version control, and it is quite interesting, not to say even confrontational, to look at those early iterations of what was to become the lower previsions book. The earliest LaTeX files I have of the book date from February 22nd 2006: the book was 110 pages long, entitled “Lower previsions for unbounded random quantities”, and Chapters 4 and 5 already contained the core material for what is now in the second part of the book. In May of 2006, at Gert’s suggestion, we made the decision to split the book into two parts: one part based on bounded gambles (the theory initiated by Peter Williams and Peter Walley), and one part on extending the theory to unbounded gambles. This made for a more consistent and self-contained book.

The decision to split the book into two parts also opened up the avenue to discuss a very wide range of new developments in the theory. And so in the next 8 years, various interesting novel results taken from the literature were added as chapters to the first part—not just our own results, but any results that we deemed useful enough for a wide range of people. The first part of Lower Previsions thus became a reference work, giving a wide overview of the key contributions (many, but certainly not all!) to the existing mathematical theory of lower previsions, at least as far as the unconditional theory goes.

One might be tempted to ask whether part one replaces Walley’s 1991 book: it certainly does not. We give a more compact theoretical development of the theory of unconditional lower previsions, and build on that to discuss a wide range of newer developments. Whilst we do discuss some philosophical issues, this is not the focus of the book: Lower Previsions is mathematical monograph. In contrast, Walley’s book goes far deeper into philosophical issues.

Concerning the conditional theory—which we do in the second part when looking at unbounded random variables as well, we depart more clearly from Walley’s approach, instead following Williams’s ideas from the mid 70s. In fact, this also fits the mathematical nature of the work: Williams is not concerned with conglomerability. My personal opinion is that the theory without conglomerability is mathematically much nicer, and in fact, mathematically also more general, as Paolo Vicig always says. Moreover, for constructivists, conglomerability might not be as compelling anyway (this is precisely De Finetti’s objection).

Anyway, we kept adding and adding, and at some point in 2011, it was clear that we simply had to stop doing this if we wanted the book ever to be published. Incomplete chapters were cut, sometimes with pain and sadness. In particular, all reference to gauges were removed. (Gauges provide an elegant abstract way to derive many results quite quickly, however we never managed to introduce them in a way that truly felt right and elegant. Some day I hope to resurrect them.) The next two years were spent on careful reviewing and refining the material—over 400 pages—to perfection.

Introduction to Imprecise Probabilities

Introduction to Imprecise Probabilities, or more briefly, ITIP as we called it pretty much from day one, started in the pub. Frank had this brilliant idea, already for quite a while, that we should spread the imprecise message, and what was obviously missing was a good introductory text on imprecise probability. Yes, there was Walley’s Statistical Reasoning with Imprecise Probabilities, but already in 2009, Walley’s book was getting out of print, with no clear plans for a reprint or second edition despite high demand, essentially due to Peter Walley’s withdrawal from the research community. Yes, Frank knew that Gert and I were working on our book. However, it was also very clear to us that our book would not be appropriate as an introductory book to give to, say, a starting PhD student or interested researcher who is merely looking to have a broad understanding of the field, perhaps looking to apply imprecise probability in some of their research.

So, in Munich, at WPMSIIP 2009, Frank bought everybody a beer, and in a nutshell we all agreed that we would write a chapter. I still fondly recall the intoxicating cheers: “to the book!”. So, everybody got their act together—some authors considerably faster than others—and wrote a chapter.

By 2011, most of the writing was done. During 2012, the chapters were reviewed and revised. In 2013, Thomas and his team in Munich spent a lot of time on making the notation consistent across the entire book, fixing references and various other details. A final revision by the authors ensued, and then essentially the book was submitted.

Lessons Learnt

The experience in getting these two books written and published has been one with great satisfaction at the end, yet some bumps in the road that perhaps could have been avoided.

One bump, common to both books, are the copy-editing and proofing stages conducted with Wiley. For the Lower Previsions book, the copy-edits were provided directly in pdf format, and only after insisting, we were given the edited LaTeX files (actually, word files that contain LaTeX code). Some copy-edits in the pdf however were not marked in the LaTeX code. For ITIP, we simply went with the pdf as it concerned far less material per author and hence not worth the hassle.

This said, I found the Wiley folks very helpful and supportive. Their operations however are not entirely adapted to support very mathematical works, where you want to stick to LaTeX for as long as possible: it is really non-trivial to indicate how, for example, you need a mathematical formula to be fixed. After Wiley’s copy-edits and our corrections to their edits, the end result was probably slightly better.

The main problem was actually punctuation: their copy-editors had a very different way to go about punctuation, and changed commas in every few sentences. It is however very easy to change the meaning of something by changing punctuation. So you really have to be very careful in checking all edits rigorously. Usually they get it right, but too often they do not.

I am unsure as to how one can really avoid these publisher bumps. The LaTeX issue is a matter of communicating clearly your expectations—then still you have to be lucky that these expectations are communicated through to the right person. The copy-editing issue is very hard to avoid: it would be quite unlikely to have a copy-editor well versed in mathematics. Perhaps they could give better guidance on punctuation and other style matters, so the copy-editing is less intrusive.

Another issue is time. How do you actually finish the book in time? This was not as much an issue for ITIP: the scope of the book was well contained and did not change throughout the process. Yes, it was perhaps a bit later than we had foreseen, but not by five years. So, if at all possible, my strong recommendation would be to make sure that indeed the scope is clear and contained, and will remain so. For Lower Previsions, we decided to move direction a few times, sometimes quite drastically, at the expense of time. That said, I think we made the right decisions. Perhaps we could have taken them earlier.

Finally, a lesson learnt from ITIP: how to edit a volume with many contributors? If you want the volume to exhibit consistency, you really need someone to go over the entire thing and make relentless edits. This is not something, say, Wiley’s copy-editor can do, due to the highly specialized field. Frank, Gert, and myself, are eternally grateful to Thomas for having taken this tremendous task upon himself.

About the author

Matthias Troffaes is senior lecturer in statistics at the Department of Mathematical Sciences, Durham
University, UK. His research interests include the foundations of statistics and decision making under severe uncertainty, with applications to engineering, renewable energy, and environment.
June 24th, 2014 by ()

Ignacio Montes’s PhD thesis on Comparison of alternatives under Uncertainty and imprecision

This thesis, supervised by Enrique Miranda and Susana Montes, was defended on May 16th. The jury was composed of Susana Díaz, Serafín Moral and Bernard De Baets.


This thesis deals with the problem of comparing alternatives defined under some lack of information, that is considered to be either uncertainty, imprecision or both together.

  •  Alternatives defined under uncertainty are modeled by means of random variables, and therefore they are compared using stochastic orders [4].
  • When the alternatives are defined under both uncertainty and imprecision, they are modeled by means of sets of random variables, and tools of the Imprecise Probability Theory [7] are used.
  • When the alternatives to be compared are defined under imprecision, but without uncertainty, they can be modeled using fuzzy sets or any of its extensions, like for instance Atanassov Intuitionistic Fuzzy Sets [1].

Comparison of random variables

Stochastic orders are methods that allow the comparison of random quantities. In this thesis we have focused on stochastic dominance and statistical preference, two stochastic orders that possess totally different interpretations. The former is based on the direct comparison of the cumulative distribution functions associated with the random variables, so it only uses marginal information, while the latter is based on a probabilistic relation providing preference degrees and using the joint distribution.

Although these stochastic orders are not related in general, we have found conditions under which stochastic dominance implies statistical preference. These conditions are based on the type of variables and the copula [5] that links them: independent random variables, simple or continuous and comonotone random variables, simple or continuous and countermonotone random variables or continuous random variables coupled by an Archimedean copula.

Both stochastic dominance and statistical preference are intended for the pairwise comparison of random variables. The reason is that stochastic dominance imposes a too strong condition for comparing three (or more) cumulative distribution functions (they must be ordered!)  and statistical preference does not prevent the existence of cycles, that is, it is not transitive [2]. For this reason we have proposed an extension of statistical preference for the comparison of more than two random variables at the same time. It preserves the same advantages than the pairwise statistical preference: it is a complete order, it provides degrees of preference, so the greater the degree the most preferred is the variable, and it uses all the available information because it is based on the joint distribution of all the random variables. Furthermore, we have shown that this general statistical preference can be applied to decision making problems with linguistic labels.

Comparison of sets of random variables

When the alternatives are defined under both uncertainty and imprecision, they are modeled by means of sets of random variables with an epistemic interpretation, meaning that all we know about the real (but unknown) random variable is that it belongs to the set.

In this framework, the first thing we have done is to extend stochastic orders to the comparison of sets of random variables instead of single ones. Any stochastic order has been extended in six different ways, and some of the choice between these extensions depends on the given interpretation (pessimistic or optimistic). We have seen that when the stochastic order to be extended is the expected utility, our six extensions are quite related to some usual criteria of the decision making with imprecise probabilities [6], such as interval dominance, maximax or maximin criteria,…

When the stochastic order to be extended is stochastic dominance, the comparison of the set of random variables is made by means of the comparison of their associated sets of cumulative distribution functions. In this sense, since each set of cumulative distribution functions can be summarized by means of its associated p-box, we have seen that there is a strong connection between the six extensions of stochastic dominance and the comparison of the bounds of the associated p-boxes by means of the stochastic dominance. Finally, when we extend statistical preference, the extensions are related to the comparison of the lower or upper medians of the adequate set of random variables.

Our general approach can be applied to three particular situations:

1)    When we want to compare belief functions, we can consider their associated credal sets, and apply there our six extensions of stochastic dominance. In this framework, our six extensions give rise to the four possibilities considered by Denouex in [3] for the comparison of belief functions with respect to stochastic dominance.

2)    When we want to compare random variables with imprecise utilities, we can consider random sets modeling our lack of information. Then, since we are using an epistemic interpretation, all we know about the real (but unknown) random variables is that they belong to the sets of measurable selections. Thus, in order to compare the random sets we can compare their sets of measurable selections by means of the extension of the adequate stochastic order.

3)    When we want to compare random variables and we have imprecise knowledge about the probability of the initial space, we can consider a credal set. In this situation, any random variable defines a set of random variables formed by the combinations of the random variable with the different probabilities in the credal set.

Next step was to investigate how to perform the comparison of random variables with imprecise joint distribution. In this situation we use p-boxes and bivariate p-boxes to model the imprecise knowledge about the marginal and joint distribution functions, respectively, and we use sets of copulas, whose information is summarized by an imprecise copula, to model the unknown copula. Sklar’s Theorem [5] is a well-known result on Probability Theory that allows expressing the joint distribution function in terms of the marginals. However, when there is imprecision about the joint (or the marginals) distribution function, it cannot be applied. For this reason, we have extended Sklar’s Theorem to an imprecise setting. This result has shown to be very useful in two situations:

1)    When we have two marginal p-boxes and we want to compute their natural extension, we can apply the imprecise Sklar Theorem to the marginal with the imprecise copula determined by Lukasiewicz’s and  minimum copulas.

2)    When we want to compute the strong product of the marginal p-boxes, we can apply the imprecise Sklar Theorem to the marginals with the product copula.

Comparison of intuitionistic fuzzy sets

Finally, when the alternatives to be compared are defined under imprecision, but without uncertainty, they can be modeled using fuzzy sets or any of its extensions, like for instance Atanassov Intuitionistic Fuzzy Sets (AIFS, for short). Several measures of comparison of this kind of sets can be found in the literature, and they are classified into two main families: distances and dissimilarities. This thesis introduces a new family of measures of comparison of AIFS: divergences, that impose stronger conditions than dissimilarities. We have also seen that these three measures can be included in a more general measure of comparison of AIFS, in the sense that depending on the required conditions, we may obtain a distance, a dissimilarity or a divergence.

A particular type of divergences, those satisfying a local property, is studied in detail, shown to have interesting properties, and applied to decision making and pattern recognition.


The main contributions of this thesis can be summarized as follows:

  • Investigation of the properties of stochastic dominance and statistical preference. In particular, study of conditions under which both are related.
  • Extension of statistical preference for the comparison of more than two random variables simultaneously.
  • Extension of stochastic orders to the comparison of sets of random variables instead of single ones.
  • Particular cases: imprecise stochastic dominance, related to the comparison of bounds of p-boxes, and imprecise statistical preference, related to the comparison of lower and upper medians.
  • The definitions introduced by Denoeux are included as particular case of our more general approach.
  • Two particular situations in decision making: comparison of random variables with imprecise utilities and beliefs.
  • Imprecise version of Sklar’s Theorem.
  • Divergences as a new measure of comparison of AIFS.
  • Local divergences applied to decision making and pattern recognition.

Basic references

[1] K. Atanassov. Intuitionistic Fuzzy Sets. Fuzzy Sets and Systems, 20:87-96, 1986.

[2] B. De Schuymer, H. De Meyer, B. De Baets. A fuzzy approach to stochastic dominance of random variables. Lecture Notes in Artificial Intelligence 2715, 253-260, 2003.

[3] T. Denoeux. Extending stochastic ordering to belief functions on the real line. Information Sciences, 179: 1362-1376.

[4] A. Müller and D. Stoyan. Comparison Methods for Stochastic Models and Risks. Wiley, 2002.

[5] R. Nelsen. An introduction to copulas. Springer, New York, 1999.

[6] M. C. M. Troffaes. Decision making under uncertainty using imprecise probabilities. International Journal of Approximate Reasoning, 45(1):17-29, 2007.

[7] P. Walley. Statistical reasoning with imprecise probabilities. Chapman and Hall, London, 1991.

About the author


Ignacio Montes started his PhD in the UNIMODE Research Unit ( in 2010 with a grant of the Spanish Ministry of Science. He is currently a member of the Dep. of Statistics and O.R of the University of Oviedo, Spain.


April 14th, 2014 by ()

Improb: A Python library for lower previsions and decision making, by Matthias C.M. Troffaes


Improb started in 2008 as a fairly small library for solving simple toy examples involving natural extension. The idea was to support exact rational calculations for lower previsions, by means of Komei Fukuda’s linear programming package cddlib. The very first incarnation of the library simply supported calculating the (unconditional) natural extension from any finite collection of assessments, and checking for avoiding sure loss and coherence.

For about two years, not much happened with the code, until 2010.

At that time, Nathan Huntley and myself were studying sequential decision problems, and we needed some way of testing our algorithms, so the library grew support for common decision rules such as maximality, Gamma-maximin, and interval dominance. It provided us with a good platform for testing all kinds of properties, and finding counterexamples with relative ease.

In that same year, the SIPTA summerschool was to be held in Durham. I was looking for a way to demonstrate some simple decision rules on some toy examples in my summerschool lecture, in order to teach the students how to apply the principles of decision making with imprecise probabilities on some of their own toy examples, and even perhaps in their own research. It seemed obvious then that improb could suit this purpose, however, the library was not really designed with “user friendliness” in mind. So, I spent about a month revamping the library, and in the end I did manage to use the library for the school. In fact, the first public version of the library, 0.1.0, was release just before the summerschool.

Much to my pleasure, some students actually managed to use it. Most notably, Erik Quaeghebeur provided valuable feedback, and even contributed to the project during the course of 2011, leading to the 0.1.1 release.

The Future of Improb?

Embarrassingly, the 0.1.1 release in 2011, almost three years ago, was also the last public release. In retrospect, there are perhaps three reasons for this.

First, although the code has been relatively untouched in the last two years, the code is actually quite stable and appears to be free of bugs. It has been tested and worked with for a long time. (Therefore, it may still provides a good starting point for anyone wanting to play around with lower previsions and decision making.)

A second reason goes back to 2011, when Erik started murasyp which has a much cleverer way of representing certain mathematical objects internally, as well as being more general in aim: murasyp allows arbitrary sets desirable gambles, not just lower previsions.

Thirdly, working with multivariate spaces in improb is a bit of a pain. Extending the library to support, say, imprecise Bayesian networks, and more general multivariate statistical reasoning, is somewhat non-trivial.

The latter two concerns have led me to believe that I need to rethink the design of improb. Some work has happened in the development branch, in particular to find a good programming interface for multivariate work, however these efforts are still not entirely satisfactory. Nevertheless, with the SIPTA summerschool coming up again this summer in Montpellier, there will be renewed reason to spend some quality time on improb.

Anyway, given improb’s relative maturity, and its suitability for solving toy problems quickly and with an easy interface, a introductory tutorial about improb seems more than worth it, hence this blog post.


First, you will want to install the library. On Linux, assuming that you have Python 2.7 and the GMP development libraries (GMP is used for exact calculations with rationals), you can simply type the following from your favorite shell:

pip install pycddlib --user
pip install improb --user

(The --user option will cause the installation to happen locally without requiring administrator rights.) On Windows, you will need Python 2.7 as well as the following packages from PyPI (pick the 2.7 .exe installers): see and


Let us get started and specify a simple lower prevision. From the Python prompt, type

from improb.lowprev.lowpoly import LowPoly
lpr = LowPoly(pspace=4)

Now, lpr will be a (initially, vacuous) lower prevision defined on a possibility space with four elements, namely Ω = {0, 1, 2, 3}. The LowPoly class is the most general class in improb for lower previsions: it can represent any conditional lower prevision with finite domain. As just mentioned, initially, lpr is vacuous. Let us verify that this is indeed the case:

x = [1, 4, 3, 2]

This code will calculate the lower and upper prevision of the gamble X defined by X(0) = 1, X(1) = 4, X(2) = 3, X(3) = 2, via natural extension of the provided assessments. Note that we denoted the gamble X by a lower case x in Python; this is merely to follow Python coding standards. Because we have not specified any assessments, lpr is vacuous, and will return 1 and 4 for the lower and upper prevision of X. As you can see, we simply used a list to specify the gamble: the first element of the list is the value of the gamble for the first element of the possibility space, and so on. 1

Natural Extension

Let us do something more interesting. Let Y(0) = Y(1) = 1 and Y(2) = Y(3) = 5. Suppose we know that the expectation of Y must lie between 2 and 3. How does this affect the bounds?

x = [1, 4, 3, 2]
y = [1, 1, 5, 5]
lpr.set_lower(y, 2)
lpr.set_upper(y, 3)

We now get 1.25 and 3.75 instead of 1 and 4 for the lower and upper prevision of X: our bounds on the expectation of Y had tightened our bounds on the expectation of X.

These kind of calculations work with any number of gambles. Improb will correct incoherent assessments as expected from natural extensions, as in for instance

from improb.lowprev.lowpoly import LowPoly
lpr = LowPoly(pspace=3)
x = [0, 2, 3]
y = [0, 4, 6]
lpr.set_lower(x, 1)
lpr.set_lower(y, 3)

Because Y = 2X, the lower prevision of Y must be twice the lower prevision of X; so lpr.get_lower(x) will return 1.5 rather than 1, reflecting the information embodied by the lower bound on the expectation on Y.

In case of conflict, as in

from improb.lowprev.lowpoly import LowPoly
lpr = LowPoly(pspace=3)
x = [0, 2, 3]
y = [0, 4, 6]
lpr.set_upper(x, 1)
lpr.set_lower(y, 3)

then improb will raise a ValueError. The conflict above arises from the fact that if the expectation of X is bounded above by 1 then the expectation of Y = 2X must be bounded above by 2: the lower bound of 3 can never be satisfied. Improb simply detected this situation for us, saving us from shooting ourselves in the foot.

There is much more to improb’s lower previsions, too much to cover in a short blog post. Suffice it to note that we can also work with conditional lower previsions (using the Walley-Pelessoni-Vicig algorithm), as well as alternative algorithms for calculating the natural extension in specific cases, for example via Mobius inversion, Choquet integration, ε-contamination, or even plain linear expectation. The documentation for improb.lowprev has all the details.

Decision Making

As mentioned in the introduction of this post, improb has been used quite extensively in the study of decision making with imprecise probabilities. Consequently, improb has reasonably good support for the standard decision rules that are used in imprecise probability, namely Γ-maximin, Γ-maximax, Hurwicz, interval dominance, and maximality. 2

How does this work in practice? We start with specifying our lower prevision.

from improb.lowprev.lowpoly import LowPoly
lpr = LowPoly(pspace=3)
x = [1, 2, 3]
lpr.set_lower(x, 1.5)
lpr.set_upper(x, 2.5)

From our lower prevision, we create an optimality operator.

from improb.decision.opt import OptLowPrevMax
opt = OptLowPrevMax(lpr)

OptLowPrevMax is the class for creating optimality operators according to maximality. There are similar classes for other optimality operators, and again, we refer to the documentation for details.

We can readily apply our optimality operator opt to any set of gambles. For example,

gambles = [[1, 2, 3], [1.5, 1.4, 1.3]]

will report that the gamble [1, 2, 3] is optimal, i.e. it dominates [1.5, 1.4, 1.3] for the given lower prevision lpr.

The object opt th