#### Contact

**Organizers:**

Dr. Pasquale Cirillo (Probability)

Dr. Jakob Söhl (Statistics)

# Seminars

## Seminar Series in Probability and Statistics

All seminars are at lunch time: feel free to bring your sandwich!

June 27, 2017: Christoph Hofer-Temmel** (NLDA)**

When: Tuesday June 27th, 12:45

Where: TBA

*TBA*

**June 20, 2017: Matthias Gorny**** (Université Paris-Sud)**

When: Tuesday June 20th, 12:45

Where: TBA

*TBA*

June 13, 2017: No Seminar

June 9, 2017: Rob Ross** (TU Delft)**

When: Friday June 9th, 12:45

Where: TBA

*TBA*

**June 6, 2017: Yining Chen**** (LSE)**

When: Tuesday June 6th, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.

*TBA*

**May 30, 2017: Stéphanie van der Pas (LUMC and Leiden University)**

When: Tuesday May 30th, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.

*TBA*

**May 23, 2017: Jere Koskela (TU Berlin)**

When: Tuesday May 23rd, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.

*Consistency results for Bayesian nonparametric inference from processes with jumps*

Consistency can informally be thought of as the ability to infer the true data generating model from a sufficiently large amount of data, and has been regarded as a minimal condition for good inference procedures for several decades. It is also notoriously difficult to verify in the Bayesian nonparametric setting. In recent years, positive results have been established for discretely observed, diffusions under restrictive but verifiable conditions on the prior. I will present an introduction to Bayesian nonparametric inference and posterior consistency, and show how these results for diffusions can be generalised to jump-diffusions under an additional identifiability assumption. Similar arguments will also be shown to yield posterior consistency for a separate class of processes called Lambda-Fleming-Viot processes: inhomogeneous, compactly supported compound Poisson processes arising as models of allele frequencies in population genetics. Identifiability can also be verified rather than assumed for Lambda-Fleming-Viot processes, which results in a tractable set of conditions for posterior consistency that is satisfied e.g. by the popular Dirichlet process mixture model prior.

**May 16, 2017: Pasquale Cirillo (TU Delft)**

When: Tuesday May 16th, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.*Exact algorithms for multinomial extremes*

The multinomial distribution is the distribution arising from throwing *n* independent balls into *k* urns. These urns may have the same attraction probability (equiprobability) or different probabilities (*J * populations case).

In this talk I will introduce and discuss brand new algorithms for the computation of the exact probabilities of multinomial extremes: maximum, minimum, range and sums of the first *m* order statistics. I will focus on the case of equiprobability, which is the most relevant from a statistical point of view, but I will also discuss how to treat the *J * populations case. I will show that all these probabilities can be obtained iteratively, using a rather intuitive urn-tree construction.

The exact algorithms, as expected, outperform the approximations available in the literature, especially for small values of *n* and *k*.

This is a joint work with Anton Ogay (TU Delft).

**May 9, 2017: Hakan Güldas (Leiden University)**

When: Tuesday May 9th, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.

*Random walks on dynamic configuration models*

In this talk, first I will introduce dynamic configuration model which is a dynamic random graph model in discrete time. Then, I will go into details of our results about mixing times of simple random walks on dynamic configuration model. The key property of the dynamic configuration model is that the degrees of the vertices do not change over time. Thanks to this property, the notion of stationary distribution for the random walk on the graph makes sense and mixing occurs although the random walk itself is not Markovian. The results I will give identify the behaviour of mixing times in terms of the proportion of edges that changes at every step of graph dynamics when the number of vertices is large.

Result are based on a joint work with Luca Avena, Remco van der Hofstad and Frank den Hollander.

**May 2, 2017: Moritz Schauer (Leiden University)**

When: Tuesday May 2nd, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.*Stochastic monotonicity of Markov processes - A generator approach*We consider stochastic orders on random variables which can be defined in terms of expectations of test functions. Notable examples are the standard stochastic order induced by the increasing functions or the convex order induced by convex functions, capturing the size and spread of random variables. In general, we consider cones of test functions characterized by Φ f ≥ 0 for some linear operator Φ.

Of particular interest are stochastically monotone Markov processes which preserve stochastic order properties in time. The semigroup {S(t): t ≥ 0} of a monotone Markov processes defined by S(t) f(x) = E [ f(X(t)) | X(0) = x] maps these cones into themselves.

We introduce a new functional analytic technique based on the generator A of the semi-group of a Markov process X(t) and its resolvent to study the property of stochastic monotonicity. We show that the existence of an operator B with positive resolvent such that Φ A - B Φ is a positive operator for a large enough class of functions implies stochastic monotonicity. This establishes a technique for proving stochastic monotonicity and preservation of order for Markov processes that can be applied in a wide range of settings including various orders for diffusion processes with or without boundary conditions and orders for discrete interacting particle systems.

Joint work with Richard C. Kraaij (Ruhr-University of Bochum)

**April 25, 2017: Maaneli Derakhshani (Utrecht University)**

When: Tuesday April 25th, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.*Hints Toward A Stochastic Hidden-Variables Foundation For Quantum Mechanics*

It is well-known that standard quantum theory is plagued by conceptual and technical problems, most notably the quantum measurement problem. The quantum measurement problem indicates that standard quantum theory (whether in non-relativistic or relativistic or quantum-gravitational form) cannot be a fundamental theory of the physical world, and must be replaced by a measurement-problem-free theory of quantum phenomena. Among the viable alternatives to standard quantum theory are nonlocal contextual 'hidden-variable' theories. In this talk, it will be shown that there are tantalizing meta-theoretical hints that the Schroedinger equation and Born-rule interpretation of the wavefunction in standard quantum mechanics have deeper foundations in some nonlocal contextual theory of stochastic hidden-variables. This will be shown by drawing surprising and little-known correspondences between the mathematical structures of Schroedinger's equation and quantum expectation values of physical observables, one the one hand, and the mathematical structures of (1) classical statistical mechanics in the Hamilton-Jacobi representation, (2) the Einstein-Smoluchowski theory of classical Brownian motion, and (3) de Broglie's famous model of a clock particle guided by phase waves, on the other. Finally, it will be suggested that Nelson's stochastic mechanics, and a recent generalization of it proposed by us, constitutes the anticipated theory of stochastic hidden-variables.

**April 18, 2017: Easter Break**

**April 11, 2017: Dutch Math Congress**

April 4, 2017: Ronald Meester (University of Amsterdam)

When: Tuesday April 4th, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.*Why the law of total probability is sometimes not desirable, and how the the theory of belief functions helps to take care of this*

We discuss some examples of betting situations in which the law of total probability fails. Since this law follows from the axioms of Kolmogorov and the definition of conditional probability, it follows that a more general theory is necessary. I will formulate more flexible axioms which turn out to characterize belief functions, a well known generalisation of probability measures. Within this theory, conditional belief functions can be defined in various ways, corresponding, roughly, to conditioning on either a necessary truth or a contingent truth. As such, the classical theory is extended and refined at the same time. I will argue that when probability is interpreted epistemically, one should always use belief functions rather than Kolmogorov probability.

This is joint work with Timber Kerkvliet.

**March 28, 2017: Gourab Ray (Cambridge University)**

When: Tuesday March 28th, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.*Universality of fluctuation in the dimer model*

The dimer model is a very popular model in statistical physics because of its exact solvability properties. I will try to convince you that the fluctuation in the dimer model is universal in the sense that it is more or less independent of the underlying graph and also the topology the graph is embedded in and is given by a form of Gaussian free field.

Joint work with Nathanael Berestycki and Benoit Laslier.

**March 21, 2017: Andrew Duncan (University of Sussex)**

When: Tuesday March 21st, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.*Measuring Sample Quality with Diffusions*

To improve the efficiency of Monte Carlo estimators, practitioners are turning to biased Markov chain Monte Carlo procedures that trade off asymptotic exactness for computational speed. While a reduction in variance due to more rapid sampling can outweigh the bias introduced, the inexactness creates new challenges for parameter selection. In particular, standard measures of sample quality, such as effective sample size, do not account for asymptotic bias. To address these challenges, we introduce a new computable quality measure based on Stein's method that quantifies the maximum discrepancy between sample and target expectations over a large class of test functions. We demonstrate this tool by comparing exact, biased, and deterministic sample sequences and illustrate applications to hyperparameter selection, convergence rate assessment, and quantifying bias-variance tradeoffs in posterior inference.

**March 14, 2017: Kolyan Ray (Leiden University)**

When: Tuesday March 14th, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.*Asymptotic equivalence between density estimation and Gaussian white noise revisited*

Asymptotic equivalence between two statistical models means that they have the same asymptotic properties with respect to all decision problems with bounded loss. A key result by Nussbaum states that nonparametric density estimation is asymptotically equivalent to a suitable Gaussian shift model, provided that the densities are smooth enough and uniformly bounded away from zero.

We study the case when the latter assumption does not hold and the density is possibly small. We further derive the optimal Le Cam distance between these models, which quantifies how close they are. As an application, we also consider Poisson intensity estimation with low count data.

This is joint work with Johannes Schmidt-Hieber.

March 7, 2017: Frank van der Meulen (TU Delft)

When: Tuesday March 7th, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.*Bayesian estimation for hypo-elliptic diffusions*

Suppose X is a discretely observed diffusion process and we wish to sample from the posterior distribution of parameters appearing in either the drift coefficient or the diffusion coefficient. As the likelihood is intractable a common approach is to derive an MCMC algorithm where the missing diffusion paths in between the observations are augmented to the state space. This requires efficient sampling of diffusion bridges. In recent years some results have appeared in the "uniformly elliptic" case, which is characterised by nondegeneracy of the covariance matrix of the noise. The "hypo-elliptic" case refers to the situation where the covariance matrix of the noise is degenerate and where observations are only made of variables that are not directly forced by white noise. As far as I am aware, not much is known how to sample bridges in this case.

In this talk I will share some recent ideas on extending earlier results with Harry van Zanten (UvA) and Moritz Schauer (Leiden), derived under the assumption of uniformly ellipticity, to this setting.

Joint work with Harry van Zanten (Uva), Moritz Schauer (Leiden) and Omiros Papaspilopoulos (Universitat Pompeu Fabra)

**February 28, 2017: No Seminar**

February 21, 2017: Pasquale Cirillo (TU Delft) - Cancelled

When: Tuesday February 21st, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.*Interacting Urn Systems and a Financial Application*

**February 9 (Extra Thursday!!!), 2017: Gareth Roberts (University of Warwick)**

When: Thursday February 9th, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.*Towards not being afraid of the big bad data set*

February 7, 2017: Nick Wormald (Monash University)

When: Tuesday February 7th, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.*A natural infection model*

Suppose that individuals are randomly placed points in space according to a Poisson process, and have two states, infected or healthy. Any infected individual passes the infection to any other at distance d according to a Poisson process, whose rate is a function f(d) of d that decreases with d. Any infected individual heals at rate 1. Initially, one individual is infected. An epidemic is said to occur when the infection lasts forever. We investigate conditions on f under which the probability of an epidemic is nonzero. This is joint work with Josep Diaz and Xavier Perez Gimenez.

**January 31, 2017: Guido Bacciagaluppi (Utrecht University)**

When: Tuesday January 31st, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.

*Quantum probability and contextuality*

In this talk, I shall introduce the generalised theory of probability that arises naturally in quantum mechanics, emphasising its understanding in terms of 'contextuality', and discussing whether and in what sense modelling such phenomena indeed requires going beyond Kolmogorovian probability.

January 24, 2017: Cancelled

**January 17, 2017: Arnaud Le Ny (Université Paris-Est Marne-la-Vallée)**

When: Tuesday January 17th, 12:45

Where: TU Delft, Faculty EWI, Mekelweg 4, Room D@ta.

*(Very) Persistent Random Walks*

In this talk, we shall describe recent works [1] and (maybe) [2] in which we investigate asymptotic properties of one dimensional very Persistent Random Walks (PRW). PRW are correlated walks whose increments are, on the contrary to simple random walks, not i.i.d. but rather dependent in a Markov (finte order) way. They have been widely studied since the mid of last century under different vocables as Goldstein-Kac, correlated or again persistent walks. Due to the extra memory induced by the increments, these random walks are not Marokov proecesses anypore. By very persistent we mean here a model in which even the increments are not Markov, but rather Variable Length Markov Chains whose conditional laws directly depend of the time already spent in the given direction. Equivalently, we are given two independent sequences of i.i.d. persistence times, in a general possibly non-summable framework that extends previous work of Malduin et al. on Directionnally Recurrent Random Walks [3]. Using an extension of Erickson's criteria [4], we provide a general classification of recurrence vs. Transience in term of drift or tail properties depending on the intial laws, and also identify different regime in the scaling limits for persistent times lying in the bassin of attraction of stable laws.

This is a joint work with P. Cénac (Dijon), B. de Loynes (Rennes) and Y. Offret (Dijon).

[1] P. Cénac, A. Le Ny, B. de Loynes, Y. Offret. Persistent Random Walks I : Recurrence vs. Transience. J. of Theo. Probab. 29, 2016/17.

[2] P. Cénac, A. Le Ny, B. de Loynes, Y. Offret. Persistent Random Walks II : Functional Limit Theorems. Preprint https://arxiv.org/abs/1612.00238.

[3] R. Malduin, M. Monticino, H. von Weisäcker. Directionally Reinforced Random Walks. Adv. In Math. 117, no 2 : 239—252, 1996.

[4] K. Erickson. The Strong Law of Large Numbers when the Mean is Undefined. Trans. Amer. Math. Soc. 185:371--381, 1973.