10th World Congress in Probability and Statistics

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

Invited 17

Approximate Bayesian Computation (Organizer: Yanan Fan)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 19 Mon, 8:30 AM — 9:00 AM EDT

Approximate inference for ordinal linear regression

Jean-Luc Dortet-Bernadet (Université de Strasbourg)

4
Ordinal regression remains one of the most useful methods for analysing data arising from ordered responses, such as those typically found in opinion surveys. We consider a flexible linear ordinal regression model which, unlike the majority of existing ordinal regression models, allows to differentiate covariate effects between different levels in the response. For scalable inference, we develop a VB approach based on truncated Gaussian distributions. A real application on data arising from student satisfaction surveys is given.

(joint work with Yanan Fan)

Generalized Bayesian likelihood-free inference using scoring rules estimators

Ritabrata Dutta (University of Warwick)

4
We propose a framework for Bayesian Likelihood-Free Inference (LFI) based on Generalized Bayesian Inference using scoring rules (SR). SR are used to evaluate probabilistic models given an observation; a proper SR is minimised in expectation when the model corresponds to the true data generating process for the observation. Use of a strictly proper SR, for which the above minimum is unique, ensures posterior consistency of our method. Further, we prove outlier robustness of our posterior for a specific SR. As the likelihood function is intractable for LFI, we employ consistent estimators of SR using model simulations in a pseudo-marginal Monte Carlo Markov chain setup; we show the target of such a chain converges to the exact SR posterior with increasing number of simulations. Furthermore, we note popular LFI techniques such as Bayesian Synthetic Likelihood (BSL) can be seen as special cases of our framework using only proper (but not strictly so) SR. We empirically validate our consistency and outlier robustness results and show how related approaches do not enjoy these properties. Practically, we use the Energy and Kernel Scores, but our general framework sets the stage for extensions with other scoring rules.

Q&A for Invited Session 17

0
This talk does not have an abstract.

Session Chair

Yanan Fan (University of New South Wales)

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Invited 20

Heavy Tailed Phenomena (Organizer: Stilian A Stoev)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 19 Mon, 8:30 AM — 9:00 AM EDT

Random linear functions of regularly varying vectors

Bikramjit Das (Singapore University of Technology and Design)

6
In various applications ranging from finance and insurance to network and environmental sciences, we encounter complex risk objects created using a combination of underlying risks which are heavy-tailed (or under certain assumptions regularly varying). A well-known result from Breiman says that the tail distribution of a product of a regularly varying random variable with another random variable remains regularly varying with the same index. We show that an extension of this result to a multivariate setting helps in quantifying a variety of extreme risks for linear combination of heavy-tailed underlying objects. In particular, we give a characterization of regular variation on sub-cones of the d-dimensional non-negative orthant, under random linear transformations. This allows us to compute probabilities of a variety of extreme events, which classical multivariate regularly varying models would report to be asymptotically negligible. Our findings are illustrated with applications to risk assessment in financial systems and reinsurance markets under a bipartite network structure. We also indicate applications of the result in computing multivariate risk measures, dimensionality reduction, and further extensions to stochastic processes.

This talk is based on joint work with Claudia Kluppelberg and Vicky Fasen-Hartmann.

Power laws and weak convergence of the Kingman coalescent

Henrik Hult (KTH Royal Institute of Technology)

5
The Kingman coalescent is an important and well studied process in population genetics modelling the ancestry of a sample of individuals. In this talk weak convergence results are presented that characterise asymptotic properties of the Kingman coalescent under parent dependent mutations, as the sample size grows to infinity. It is shown that the sampling probability satisfies a power-law and derive the asymptotic behaviour of transition probabilities of the block counting jump chain. For the normalised jump chain and number of mutations between types a limiting process is derived consisting of a deterministic component, describing the limit of the block counting jump chain, and independent Poisson processes with state-dependent intensities, exploding at the origin, describing the limit of the number of mutations. Finally, the results are extended to characterise the asymptotic performance of popular importance sampling algorithms, such as the Griffiths-Tavare algorithm and the Stephens-Donnelly algorithm.

This is joint work with Martina Favero.

Limit theorems for topological invariants of extreme sample cloud

Takashi Owada (Purdue University)

6
The main objective of this work is to study the topological crackle from the viewpoints of Topological Data Analysis (TDA) and Extreme Value Theory. TDA is a growing research area that broadly refers to the analysis of high-dimensional datasets, the main goal of which is to extract robust topological information from datasets. Topological crackle typically appears in the statistical manifold learning problem, referring to the layered structure of homological cycles generated by “noisy” samples, where the underlying distribution has a heavy tail. We establish various limit theorems (e.g., central limit theorems, strong laws of large numbers) for topological objects, including Betti numbers — a basic quantifier of homological cycles, and the Euler characteristics.

Q&A for Invited Session 20

0
This talk does not have an abstract.

Session Chair

Stilian A Stoev (University of Michigan, Ann Arbor)

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Invited 33

Integrable Probability (Organizer: Tomohiro Sasamoto)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 19 Mon, 8:30 AM — 9:00 AM EDT

Reversing nonequilibrium systems

Leonid Petrov (University of Virginia)

6
A typical stochastic particle model for nonequilibrium thermodynamics starts from a densely packed initial configuration, and evolves by emanating particles into the “rarefaction fan”. Imagine having air and vacuum in two halves of a room, and removing the separating barrier. I will explain how for very special (integrable) stochastic particle systems one can explicitly “undo” the rarefaction, and construct another Markov chain which “puts the air back into its half of the room”. I will also discuss the corresponding stationary processes preserving each time-t nonequilibrium measure.

Mapping KPZ models to free fermions at positive temperature

Takashi Imamura (Chiba University)

4
We find a direct connection between solvable models in the Kardar-Parisi-Zhang (KPZ) universality class and free fermonic models at positive temperature. In studies of integrable probability during the last decade, Fredholm determinant formulas have been obtained for the one dimensional KPZ equation and its integrable discretized models. However it has been a long standing problem to understand the origin of such determinantal structures. Although the final formulas are very simple, they are usually found through complicated calculations. The situation was quite different around 2000. Using the Robinson-Shensted-Knuth (RSK) algorithm Johansson showed that the current of the totally asymmetric simple exclusion process (TASEP) corresponds to a marginal of a free fermionic system at zero temperature described by the Schur measure. In this talk, we will show that there exist more general connections between the discretized models of the KPZ equations and the free fermions. On the KPZ side the models we consider are solvable one parameter deformation of the TASEP. On the free fermionic side, such deformations in fact correspond to bringing the system to positive temperature. The connections we find is enabled by a new fundamental identity between marginals of the q-Whittaker measures and the periodic Schur measures. It is obtained by comparing the Fredholm determinant formulas for the q-Whittaker measures by Imamura-Sasamoto(2019) and for the Periodic Schur measures by Borodin (2007). We will also report briefly further insights into this topic. One is the bijective combinatorics approach to the identity. There are deep mathematical structures behind it. The other is about applications of this approach to a few KPZ models. Also reported are deformations to this identity connecting the KPZ models in half spaces with Pfaffian point process. Details of these two topics will be explained by Matteo Mucciconi and Tomohiro Sasamoto respectively.

Relaxation time limit of TASEP on a ring

Jinho Baik (University of Michigan)

7
TASEP, totally asymmetric simple exclusion process, is a standard example of an interacting particle system that belongs to the KPZ (Kadar-Parisi-Zhang) universality class. The 2-dimensional random field defined by the height fluctuations of the TASEP on an infinite line converges to an universal random field, the KPZ fixed point. In this talk we discuss what happens if the space is changed to a ring and the ring size grows large together with the time in a certain critical way so that all particles are critically correlated. This talk is based on a joint work with Zhipeng Liu.

Q&A for Invited Session 33

0
This talk does not have an abstract.

Session Chair

Tomohiro Sasamoto (Chiba University)

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