Category: new presentation

Conference paper, presentation, and poster: Computable randomness is inherently imprecise

Computable randomness is inherently imprecise
Gert de Cooman and Jasper De Bock
Proceedings of Machine Learning Research, vol. 62, pp. 133–144, 2017

A conference version of one of the papers I am perhaps most proud of. Research for this paper started with discussions between Philip Dawid and myself about what prequential interval forecasting would look like, during a joint stay at Durham University in late 2014. Jasper who joined in late 2015, and I wrote an early prequential version of the present paper during a joint research visit to the Universities of Strathclyde and Durham in May 2016, trying to extend results by Volodya Vovk to make them allow for interval forecasts. In an email exchange, Volodya pointed out a number of difficulties with our approach, which we were able to resolve by letting go of its prequential emphasis, at least for the time being. This was done during research visits of mine to Jasper at IDSIA in late 2016 and early 2017.

Abstract: We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define a notion of computable randomness associated with interval, rather than precise, forecasting systems, and study its properties. The richer mathematical structure that thus arises lets us better understand and place existing results for the precise limit. When we focus on constant interval forecasts, we find that every infinite sequence of zeroes and ones has an associated filter of intervals with respect to which it is computably random. It may happen that none of these intervals is precise, which justifies the title of this paper. We illustrate this by showing that computable randomness associated with non-stationary precise forecasting systems can be captured by a stationary interval forecast, which must then be less precise: a gain in model simplicity is thus paid for by a loss in precision.

This paper was presented at the ISIPTA 2017 symposium in Lugano, Switzerland. Besides the actual paper, you can also download our presentation, and our poster.


Predictive inference under exchangeability and the Imprecise Dirichlet Multinomial Model

Invited plenary lecture (with video) at the 12th Brazilian Meeting on Bayesian Statistics (EBEB 2014) in Atibaia SP, Brazil on 10 March 2014.

This talk (and the extremely long paper to go with it) marks the happy end of a spontaneous and informal research project that started in 2004, branched off in a surprising number of directions, and taught me so much about mathematics in general, and the foundations of probability theory in particular. Along the way, I needed all the help I could get, and got it from Enrique Miranda, Erik Quaeghebeur, Jasper De Bock and Márcio Diniz.

Here’s the abstract: Coherent reasoning under uncertainty can be represented in a very general manner by coherent sets of desirable gambles. In this framework, and for a given finite category set, coherent predictive inference under exchangeability is 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 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 inference systems corresponding to (a modified version of) Walley and Bernard’s imprecise Dirichlet multinomial models (IDMMs) and the Haldane inference system.

A game-theoretic ergodic theorem for imprecise Markov chains

Invited plenary lecture at the Fifth Workshop on Game-Theoretic Probability and Related Topics (GTP2014), held in Guanajuato, Mexico, 10-13 November 2014.

Based on research I’ve done in the past year with Jasper De Bock and Stavros Lopatatzidis. I had been trying to prove an ergodic theorem in an imprecise probabilities context ever since I started work in the field. These things take time and patience, especially in imprecise probabilities, because many of the tools present in the precise counterpart are simply not available yet. Or take surprisingly different forms.

Here’s the abstract: We prove a game-theoretic version of the strong law of large numbers for submartingale differences, and use this to derive a pointwise ergodic theorem for discrete-time Markov chains with finite state sets, when the transition probabilities are imprecise, in the sense that they are only known to belong to some convex closed set of probability measures.

Lecture at the SIPTA School 2014 in Montpellier

I have just finished lecturing at the SIPTA 2014 School in Montpellier, the 6th installment of a series of summer schools on imprecise probabilities, held at the beautiful location of the Institut de Botanique.

The "salle de conférences" at the Institut de Botanique in Montpellier

The “salle des actes” at the Institut de Botanique in Montpellier

Erik Quaeghebeur and I presented a joint lecture on inference. You can download our slides for the joint presentation, lecture notes for Erik’s general discussion of inference and his additional examples for binomial sampling, and lecture notes for my discussion of symmetry and exchangeability.

SIPTA School 2014 Participants

SIPTA School 2014 Participants

Inference using sets of desirable gambles

Invited plenary lecture at the 12th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2013) in Utrecht, The Netherlands on 10 July 2013.

Voordracht voor TechBoost: Waarschijnlijkheid (en) modelleren

Op donderdag 18 april gaf ik in Auditorium Magnel (Campus Ardoyen, Zwijnaarde) een voordracht, of eigenlijk een les, in het kader van TechBoost, een initiatief van AIG.

Ik heb genoten van de unieke sfeer die een geïnteresseerd publiek creëert, en dat kun je zien op de foto’s die het AIG heeft vrijgegeven. Ook mijn slides zijn voor consultatie beschikbaar.

Mocht je vragen hebben, dan contacteer je mij het beste via e-mail.

A link between Game-Theoretic Probability and Imprecise Probabilities

Lecture at the Fourth Workshop on Game-Theoretic Probability and Related Topics (GTP2012), held in Tokyo, Japan, 12-14 November 2012.

Based on research I’ve done during the past few years (and for the last part in particular during the last few months) with Filip Hermans, Jasper De Bock and Enrique Miranda.

Here’s the abstract: In game-theoretic probability (GTP) there is a fundamental formula (which we will call the Shafer-Vovk-Ville, or SVV, formula) for expressing lower and upper prices for a variable defined on the terminal situations of an event tree associated with a game. In GTP it is used as a given: a starting point for much of the development of the theory. In earlier work, we have shown how, for event trees that are bounded, this formula can be derived on a behavioral approach and with a different interpretation, in the context of the theory of imprecise probabilities (IP), from two rationality requirements: coherence and cut conglomerability. In the present talk, we discuss how something similar, but more involved, can also be done for unbounded event trees: besides coherence, we impose two additional rationality axioms, bounded cut conglomerability and bounded cut continuity. Interestingly, our approach shows that in deriving the SVV formula, two types of infinity have a part, and can be treated separately: bounded cut conglomerability tries to cope with infinity in the width of the event tree, and bounded cut continuity with infinity in its depth. These additional requirements only need to be invoked when going from local to global modes: it concerns the global uncertainty models in the tree—the uncertainty about paths—, whereas for the local models—the uncertainty about the next move in a situation—we only need to impose coherence, nothing more. We explore a number of aspects and consequences of this connection between GTP and IP.

Recent developments in imprecise probabilities and probabilistic graphical models

Invited talk in the Frontiers in AI track at ECAI 2012, held in Montpellier, France, 27-31 August 2012.

Credal trees under irrelevance

Lecture at the Fifth SIPTA School on Imprecise Probabilities, held in Pescara, Italy, 16-20 July 2012.

Irrelevance, independence and coherence

Lecture at the Fifth SIPTA School on Imprecise Probabilities, held in Pescara, Italy, 16-20 July 2012.

Desirable symmetry: a few open questions about desirability and symmetry

In early November, there was a Workshop on Geometry of Imprecise Probability and related Statistical Methods (GEOMIP-11) at Durham University. I gave a talk there about some geometrical aspects of modeling symmetry in imprecise probabilities, using the sets of desirable gambles model: I used the example of finite exchangeability to identify a number of interesting research problems in this area.

New developments in credal networks

I am about to cough and sneeze myself through an invited talk (here’s the abstract) at the Statistische Woche, a five-day event organized in Leipzig, Germany to celebrate the 100th birthday of the Deutsche Statistische Gesellschaft.

Imprecise probabilities in stochastic processes and probabilistic graphical models: New developments

I have just finished giving a plenary talk at NLMUA 2011, the first international conference on Nonlinear Mathematics for Uncertainty and Its Applications, held in Beijing. My aim was to convince people that recent advances in imprecise probabilities could lead to an interesting approach to stochastic processes using imprecise probability models, and that some of the underlying ideas are already being used to good advantage in imprecise Markov chains, credal networks, and in particular imprecise Hidden Markov models.

Exchangeability: how Bruno de Finetti’s ideas thrive in indeterminate soil

Here is the PDF-file for the plenary lecture I gave on 26 July 2011 during the ISIPTA’11 session devoted to Bruno de Finetti.

Later that same day, all ISIPTA participants went to the house in Innsbruck where de Finetti was born, to unveil a plaque (photo by Inés Couso, cropped):

Plaque at Bruno de Finetti's Geburtshaus in Innsbruck

Lectures at the fourth SIPTA School on Imprecise Probability

I am currently in Durham (UK), attending the fourth SIPTA School on Imprecise probabilities, and also lecturing there on the topic of structural judgements. This is a fancy term for judgements of irrelevance, independence, or symmetry (and therefore in particular exchangeability).

It is turning out to be an interesting event, very well organised by Matthias Troffaes, with innovative lectures on various topics relevant to the field (see the website of the school for more details and pdfs of the presentations).

Erik Quaeghebeur gave a(n in many ways) beautiful talk on desirability, so if you have any interest in the notion of desirability, or in good mathematical presentations, for that matter, go and have a look.