Eindhoven SPOR Seminar

Recursive distributional equations (RDE) appear in various areas of probability such as the probabilistic analysis of algorithms, probabilistic number theory, stochastic geometry or, more recently, in distributional reinforcement learning (DRL). In this talk more applied aspects of RDE are discussed in the context of their applications. Some of the new results discussed on DRL are joint work with Julian Gerstenberg and Denis Spiegel

Investigating a cluster of deaths on a hospital ward is a difficult task for medical investigators, police, and courts. Patients do die in hospitals (the three most common causes of deaths in a hospital are, in order: cancer, heart disease, medical errors). Often such cases come to the attention of police investigators for two reasons: gossip about a particular nurse is circulating just as a couple of unexpected and disturbing events occur. Hospital investigators see a pattern and call in the police.

I will discuss two such cases with which I have been intensively involved. The first one is the case of the Dutch nurse Lucia de Berk. Arrested in 2001, convicted by a succession of courts up to the supreme court by 2005, after which a long fight started to get her a re-trial. She was completely exonerated in 2010. The second case is that of the English nurse Lucy Letby. Arrested in 2018, 2019 and 2020 for murders taking place in 2015 and 2016. Her trial started in 2022 and concluded with a “full life” sentence a couple of months ago.

There are many similarities between the two cases, but also a couple of disturbing differences. One difference being that Lucy Letby’s lawyers seem to have made no attempt whatsoever to defend her. Another difference is that statistics was used against Lucia de Berk but not, apparently, against Lucy Letby. But appearances are not always what they seem.

Report published by Royal Statistical Society on statistical issues in these cases
https://rss.org.uk/news-publication/news-publications/2022/section-group-reports/rss-publishes-report-on-dealing-with-uncertainty-i/

News feature in “Science” about myself and my work
https://www.science.org/content/article/unlucky-numbers-fighting-murder-convictions-rest-shoddy-stats

There is a current trend on reevaluating artificial intelligence (AI), its advancements and their implications to society. Uncertainty treatment plays a major role in this discussion. This talk will hopefully convince you that we can make AI more reliable and trustworthy by a sound treatment of uncertainty. Uncertainty is often modelled by probabilities, while it has been argued that some broadening of probability theory is required for a more convincing treatment, as one may not always be able to provide a reliable probability for every situation. Credal models generalize probability theory to allow for partial probability specifications and are arguably a good direction to follow when information is scarce, vague, and/or conflicting. We will present and discuss credal approaches from simple examples to sophisticated credal machine learning models and even their reach into both adversarial and causal inferences. The talk argues that we must continue to push AI forward by investing in Cautious AI.

Via a new probabilistic result, we provide (1+o(1))-asymptotics for the number of integer sequences n-1>= d_1 >= … >= d_n >= 0 that form the degree sequence of an n-vertex graph (improving both the upper and lower bound by a multiplicative n^{1/4}-factor). In particular, we determine the asymptotic probability that the integral of a (lazy) simple symmetric random walk bridge remains non-negative. This talk will explain how this problem arose, what the connection is with the problem about random walks (including what all the words in this abstract mean) and then provide a short sketch of the proof. This is based on joint work with Paul Balister, Serte Donderwinkel, Tom Johnston and Alex Scott.

Over the past 20 years, the Santa Claus problem has been the source of many beautiful algorithmic ideas and to date remains an active and fruitful research area. The problem takes its metaphorical name from the narrative of Santa Claus distributing his gifts to children in a way that the least happy child is as happy as possible. I will give an overview over the landscape of techniques used to approach the problem, previous results as well as open questions. Then I will focus on recent joint work with Bamas [STOC’23], where we make progress towards sublogarithmic approximations.

We consider a finite horizon online resource allocation/matching problem where the goal of the decision maker is to combine resources (from a finite set of resource types) into feasible configurations. Each configuration is specified by the number of resources consumed of each type and a reward. The resources are further subdivided into three types – offline, online-queueable (which arrive online and can be stored in a buffer), and online-nonqueueable (which arrive online and must be matched on arrival or lost). We prove the efficacy of a simple greedy algorithm when the corresponding static planning linear program (SPP) exhibits a non-degeneracy condition called the general position gap (GPG). In particular we prove that, (i) our greedy algorithm gets bounded any-time regret when no configuration contains both an online-queueable and an online-nonqueueable resource, and (ii) O(log t) expected any-time regret otherwise (we also prove a matching lower bound). By considering the three types of resources, our matching framework encompasses several well-studied problems such as dynamic multi-sided matching, network revenue management, online stochastic packing, and multiclass queueing systems.

It has been well-known since Aldous’ seminal work in the 90s that a critical Galton-Watson tree conditioned to survive until generation n converges, as n goes to infinity, to a continuous random tree (CRT). This is the equivalent, in the world of random trees, of the fact that a random walk with finite-variance i.i.d. increments converges to the Brownian motion. In this is joint work with Guillaume Conchon-Kerjan and Daniel Kious, we prove that a Galton-Watson tree “in random environment” (GWRE) also converges to the CRT. GWREs are GW trees in which the offspring distribution of an individual depends on its generation, and the sequence of offspring distributions (indexed by the generation) is sampled in an i.i.d. way. I will review known results on GWREs before stating our main result and giving ideas of the proof.

During this seminar, two postdocs within our cluster will introduce themselves and their research.

We discuss recent developments in the use of interacting particles for solving high-dimensional optimization problems, and provide insights into analytical guarantees for convergence in the particle mean-field limit.

Symbolic Data Analysis gives a new way of thinking in Data Science by extending the standard variables to take into consideration new kinds of data, called “symbolic”, as they cannot be reduced to numbers without losing much information. Interval data are a prime example of symbolic data. There have been a series of proposed adaptations of the Principal Component Analysis method for interval-valued symbolic data, all of which have the downside of having intermediate steps that deal with conventional data. In this talk, we use algebraic structures on the intervals, to define symbolic principal components as linear combinations of the intervals that maximise the symbolic variance. This framework provides the mathematical tools needed to use the symbolic principal components to transform the original data in a way that is mathematically coherent with the remainder of the framework and defines the principal components as solutions to maximisation problems, like what is done in conventional Principal Component Analysis. As in the conventional case, our formulation of interval-based principal component analysis can be seen as a dimensionality reduction method, since we explicitly project the original symbolic observations on the reduced space spanned by the first principal components. We conclude by exploring real-world data from the telecommunications sector, in an attempt to detect Internet redirection attacks in real time. Joint work with Rodrigo Serrão Girão and Lina Oliveira

Multivariate extreme value distributions are a common choice for modelling multivariate extremes. In high dimensions, however, the construction of flexible and parsimonious models is challenging. We propose to combine bivariate extreme value distributions into a Markov random field with respect to a tree. Although in general not an extreme value distribution itself, this Markov tree is attracted by a multivariate extreme value distribution. The latter serves as a tree-based approximation to an unknown extreme value distribution with the given bivariate distributions as margins. Given data, we learn an appropriate tree structure by Prim’s algorithm with estimated pairwise upper tail dependence coefficients or Kendall’s tau values as edge weights. The distributions of pairs of connected variables can be fitted in various ways. The resulting tree-structured extreme value distribution allows for inference on rare event probabilities, as illustrated on river discharge data from the upper Danube basin. Based on joint work with Shuang Hu and Zuoxiang Peng.

During this seminar, three postdocs within our cluster will introduce themselves and their research.