10th World Congress in Probability and Statistics

Organized Contributed Session (live Q&A at Track 2, 9:30PM KST)

Organized 08

Rough Path Theory (Organizer: Ilya Chevyrev)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 22 Thu, 8:30 AM — 9:00 AM EDT

Rough path theory and the stochastic Loewner equation

Vlad Margarint (New York University Shanghai)

5
In this talk, I will give an overview of some work carried at the intersection of Rough Path Theory and Schramm-Loewner Evolutions (SLE) Theory. Specifically, I will cover a study of the Loewner Differential Equation using Rough Path techniques (and beyond). The Loewner Differential Equation describes the evolution of a family of conformal maps. We rephrase this in terms of (Singular) Rough Differential Equations. In this context, it is natural to study questions on the stability, and approximations of solutions of this equation. First, I will present a result on continuity of the dynamics and related objects in a natural parameter that appears in the problem. The first approach will be based on Rough Path Theory, and a second approach will be based on a constructive method of independent interest: the square-root interpolation of the Brownian driver of the Loewner Differential Equation. In the second part, if time permits, I will present a result on the asymptotic radius of convergence of the Stochastic Taylor approximation of the Loewner Differential Equation and numerical simulations of the SLE trace using a novel numerical method: Ninomiya-Victoir (or Strang) splitting.

The first part is based on joint work with Dmitry Belyaev, Terry Lyons, and the second part on a collaboration with James Foster.

Rough path with jumps and its application in homogenization

Huilin Zhang (Fudan University)

6
In this talk I will present recent progress on rough path theory with jumps, consisting of Ito/forward theory and the Stratonovich/Marcus theory. It allows us to handle stochastic differential equations driven by jump noise. As an application, we show how rough paths can be applied in the homogenization of a fast-slow system proposed by Melbourne and Stuart.

This talk is based on works with Chevyrev, Friz, Korepanov and Melbourne.

Probabilistic rough paths

William Salkeld (Universite Cote d'Azur)

4
In this talk, I will explain some of the foundation results for a new regularity structure developed to study interactive systems of equations and their mean-field limits. At the heart of this solution theory is a Taylor expansion using the so called Lions measure derivative. This quantifies infinitesimal perturbations of probability measures induced by infinitesimal variations in a linear space of random variable.

This talk is based on preprints and ongoing work with my supervisor Francois Delarue at Universite Cote d'Azur.

Transport and continuity equations with (very) rough noise

Nikolas Tapia (Weierstrass Institute / Technische Universität Berlin)

4
We study the solution theory of linear transport equations driven with rough multiplicative noise. We show existence and uniqueness for rough flows driven by an arbitrary geometric rough path, and obtain a rough version of the classical method of characteristics, under a boundedness condition for the vector fields. We also obtain an adjoint RDE for the derivatives of the induced flow. Dually, we show existence and uniqueness for the associated continuity equation.

Rough walks in random environment

Tal Orenshtein (Technische Universität Berlin, Weierstrass Institute for Applied Analysis and Stochastics)

5
Random walk in random environment (RWRE) is a model to describe propagation of heat or diffusion of matter through a highly irregular medium. The latter is expressed locally in the model in terms of a random environment according to which the process evolves randomly in time. In a few fundamental classes the phenomenon of homogenization of the media takes place. One way this can be expressed is in the fact that on large scales, the RWRE fluctuates as a Brownian motion with a deterministic covariance matrix given in terms the (law of the) environment. Rough path theory enables the construction of solutions to SDEs so that the solution map is continuous with respect to the noise. One important application guarantees that if the approximation converges to the noise in the rough path topology, the SDEs driven by the noise approximations converge, in an appropriate sense, to a well-defined SDE which is different than the original one, so that the correction term is explicit in terms of the noise approximation. In this talk we shall present our current program, in which one lifts RWRE in various classes to the rough path space and shows a convergence to an enhanced Brownian motion in the rough path topology. Interestingly, the limiting second level of the lifted RWRE may have a linear correction, called area anomaly, which we identify. Except for the immediate application to approximations of SDEs, and potentially to SPDEs, this adds some new information on the RWRE limiting path. Time permitted, we shall elaborate on the tools to tackle these problems.

Based on joint works with Olga Lopusanschi, with Jean-Dominique Deuschel and Nicolas Perkowski and with Johaness Bäumler, Noam Berger and Martin Slowik.

Q&A for Organized Contributed Session 08

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This talk does not have an abstract.

Session Chair

Ilya Chevyrev (University of Oxford)

Organized 21

Recent Advances in Statistics (Organizer: Yunjin Choi)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 22 Thu, 8:30 AM — 9:00 AM EDT

Identifiability of additive noise models using conditional variances

Gunwoong Park (University of Seoul)

5
The identifiability assumption of structural equation models (SEMs) is considered in which each variable is determined by a arbitrary function of its parents plus an independent error. It has been shown that linear Gaussian structural equation models are fully identifiable if all error variances are the same or known. Hence, we prove the identifiability of SEMs with both homogeneous and heterogeneous unknown error variances. Our new identifiability assumption exploits not only error variances, but edge weights; hence, it is strictly milder than prior work on the identifiability result. We further provide a statistically consistent and computationally feasible learning algorithm. We verify through simulations that the proposed algorithm is statistically consistent and computationally feasible in the high-dimensional settings, and performs well compared to state-of-the-art US, GDS, LISTEN, PC, and GES algorithms. We also demonstrate through real human cell signalling and mathematics exam data that our algorithm is well-suited to estimating DAG models for multivariate data in comparison to other methods used for continuous data.

Multivariate functional group sparse regression: functional predictor selection

Jun Song (University of North Carolina at Charlotte)

4
In this talk, I will present a method for functional predictor selection and the estimation of smooth functional coefficients simultaneously in a scalar-on-function regression problem under a high-dimensional multivariate functional data setting. In particular, we develop two methods for functional group-sparse regression under a generic Hilbert space of infinite dimension. Then we show the convergence of algorithms and the consistency of the estimation and selection under infinite-dimensional Hilbert spaces. Simulation and fMRI data application will be presented at the end to show the effectiveness of the methods in both the selection and estimation of functional coefficients.

Causal foundations for fair and responsible machine learning

Joshua Loftus (London School of Economics)

6
Recently the social impacts of new data and information technologies started receiving more attention from scholars and the public. Many are concerned that algorithmic decision systems using machine learning or "artificial intelligence" may affect people negatively, especially in ways that reinforce harmful patterns in historic data related to attributes like race, gender, and other ethically or legally important attributes. This talk will briefly survey recent work in the field and then focus on causal modeling as a pathway beyond impossibility results and toward consensus.

Network change point detection

Yi Yu (University of Warwick)

5
This talk will be on three different projects, parametric network change point detection, nonparametric network change point detection and online network change point detection. I will provide theoretically-justified and computationally-efficient change point estimators in these three different scenarios, and discuss the theoretical difficulties in all settings.

Q&A for Organized Contributed Session 21

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This talk does not have an abstract.

Session Chair

Yunjin Choi (University of Seoul)

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