Applied Probability and Risk Seminar

The Applied Probability and Risk Seminar (APR) is the joint seminar between IEOR, theStatistics Department and the Center for Applied Probability (CAP).

Fall 2018 Seminars

Tze Lai (Stanford) | 9/13/18 | 4:10pm to 5:00pm

Speaker: Tze Lai (Stanford)
Date: Thursday, September 13, 2018
Time: 4:10pm to 5:00pm
Location: 903 SSW

Title: MCMC with Sequential State Substitutions: Theory and Applications

Abstract:

Motivated by applications to adaptive filtering that involves joint parameter and state estimation in hidden Markov models, we describe a new approach to MCMC, which uses sequential state substitutions for its Metropolis-Hastings-type transitions. The basic idea is to approximate the target distribution by the empirical distribution of N representative atoms, chosen sequentially by an MCMC scheme so that the empirical distribution converges weakly to the target distribution as the number K of iterations approaches infinity. Making use of coupling arguments and bounds on the total variation norm of the signed measure defined by the difference between the target distribution and the empirical measure induced by the sample paths of the MCMC scheme, we develop its asymptotic theory. In particular, we prove the asymptotic normality (as both K and N become infinite) of the estimates of functionals of the target distribution using the new MCMC method, provide consistent estimates of their standard errors, and derive oracle properties that prove their asymptotic optimality. Implementation details and applications, particularly to adaptive particle filtering with consistent standard error estimates, are also given.

Zachary Feinstein (Washington U in St. Louis) | 9/20/2018 | 1:10pm to 2:00p

Speaker: Zachary Feinstein (Washington U in St. Louis)
Date: Thursday, September 20, 2018
Time: 1:10pm to 2:00pm
Location: MUDD 303
Title: Pricing debt in an Eisenberg-Noe network under comonotonic endowments

Abstact:

In this talk we present formulas for the pricing of debt and equity of firms in a financial network under comonotonic endowments. We demonstrate that the comonotonic setting provides a lower bound to the price of debt under Eisenberg-Noe financial networks with consistent marginal endowments. Such financial networks encode the interconnection of firms through debt claims. The proposed pricing formulas consider the realized, endogenous, recovery rate on debt claims. Special consideration will be given to the setting in which firms only invest in a risk-free bond and a common risky asset following a geometric Brownian motion.

Yuval Peres (Microsoft Research) | 9/27/18 | 1:10pm to 2:00pm

Speaker: Yuval Peres (Microsoft Research)
Date: Thursday, September 27, 2018
Time: 1:10pm to 2:00pm
Location: MUDD 303

Title: Trace reconstruction for the deletion channel

Abstract:

In the trace reconstruction problem (arising in DNA storage) an unknown string x of n bits is observed through the deletion channel, which deletes each bit with probability q, yielding a contracted string. How many independent outputs (traces) of the deletion channel are needed to reconstruct x with high probability? This is arguably the most natural problem with such a large gap between the upper and lower bounds. The best lower bound known is of order $n^{1.25}$. Until 2016, the best upper bound was exponential in the square root of n. With Fedor Nazarov we improved the square root to a cube root (STOC 2017; also obtained by De, O’Donnell and Servedio). Classical complex analysis makes a surprise appearance in the proof. However, If the string x is random, then a sub-polynomial number of traces suffices (Joint work with Alex Zhai, FOCS 2017, for q<1/2; and with Nina Holden and Robin Pemantle , COLT 2018, for general q).

Miklos Racz (Princeton) | 10/4/2018 | 1:10pm to 2:00pm

Speaker: Miklos Racz (Princeton)
Date: Thursday, October 4, 2018
Time: 1:10pm to 2:00pm
Location: MUDD 303

Title: High-dimensional random geometric graphs

Abstract: I will talk about two natural random geometric graph models, where connections between vertices depend on distances between latent d-dimensional labels. We are particularly interested in the high-dimensional case when d is large. We study a basic hypothesis testing problem: can we distinguish a random geometric graph from an Erdos-Renyi random graph (which has no geometry)? We show that there exists a computationally efficient procedure which is almost optimal (in an information-theoretic sense). The proofs will highlight new graph statistics as well as connections to random matrices. This is based on joint work with Sebastien Bubeck, Jian Ding, Ronen Eldan, and Jacob Richey.

Nicolas Garcia Trillos (U Wisconsin)| 10/18/2018 | 1:10pm to 2:00pm

Speaker: Nicolas Garcia Trillos (U Wisconsin)
Date: Thursday, October 18, 2018
Time: 1:10pm to 2:00pm
Location: MUDD 303

Title: Large sample asymptotics of graph-based methods in machine learning: mathematical analysis and implications.

Abstract: Many machine learning procedures aimed to extract information from data can be defined as precise mathematical objects that are constructed in terms of the data. It is often assumed that the data is “big” in complexity but also in quantity, and in this “large amount of data’’ setting, a basic mathematical concept that one can explore is that of closure of a given class of statistical procedures (i.e. what are the limiting procedures as the number of data points available goes to infinity.) In this talk, I will explore this notion in the context of graph-based methods. Examples of such methods include minimization of Cheeger cuts, spectral clustering, and graph-based bayesian semi-supervised learning, among others. I will introduce some of the mathematical ideas needed for the analysis, as well as show some of the implications of it: our results show statistical consistency of the methods, provide with quantitative information in the form of scaling of parameters and rates of convergence, imply qualitative properties at the discrete level, and suggest the use of appropriate algorithms.

Philip Ernst (Rice) | 10/25/2018 | 1:10pm to 2:00pm

Speaker: NPhilip Ernst (Rice University)
Date: Thursday, October 25, 2018
Time: 1:10pm to 2:00pm
Location: MUDD 303

Title: Yule’s “Nonsense Correlation” Solved!

Abstract:

In this talk, I will discuss how I recently resolved a longstanding open statistical problem. The problem, formulated by the British statistician Udny Yule in 1926, is to mathematically prove Yule’s 1926 empirical finding of “nonsense correlation.” We solve the problem by analytically determining the second moment of the empirical correlation coefficient of two independent Wiener processes. Using tools from Fredholm integral equation theory, we calculate the second moment of the empirical correlation to obtain a value for the standard deviation of the empirical correlation of nearly .5. The “nonsense” correlation, which we call “volatile” correlation, is volatile in the sense that its distribution is heavily dispersed and is frequently large in absolute value. It is induced because each Wiener process is “self-correlated” in time. This is because a Wiener process is an integral of pure noise and thus its values at different time points are correlated. In addition to providing an explicit formula for the second moment of the empirical correlation, we offer implicit formulas for higher moments of the empirical correlation. The paper appeared in The Annals of Statistics and can be found at https://projecteuclid.org/euclid.aos/1498636874

Mark Brown(Columbia) | 10/29/2018 | 4:10pm to 5:00pm

Joint with the Statistics Seminar

Speaker: Mark Brown(Columbia)
Date: Monday, October 29, 2018
Time: 4:10pm to 5:00pm
Location: 903 SSW

Title: Taylor’s Law via Ratios, for Some Distributions with Infinite Mean

Abstract: Taylor’s law (TL) originated as an empirical pattern in ecology. In many sets of samples of population density, the variance of each sample was approximately proportional to a power of the mean of that sample. In a family of nonnegative random variables, TL asserts that the population variance is proportional to a power of the population mean. TL, sometimes called fluctuation scaling, holds widely in physics, ecology, finance, demography, epidemiology, and other sciences, and characterizes many classical probability distributions and stochastic processes such as branching processes and birth-and-death processes. We demonstrate analytically for the first time that a version of TL holds for a class of distributions with infinite mean. These distributions and the associated TL differ qualitatively from those of light-tailed distributions. Our results employ and contribute to methodology of Albrecher and Teugels (2006) and Albrecher, Ladoucette and Teugels (2010). This work opens a new domain of investigation for generalizations of TL. This work is joint with Professors Joel Cohen and Victor de la Pena.

Dan Pirjol (JP Morgan) | 11/1/2018 | 1:10pm to 2:00pm

Speaker: Dan Pirjol (JP Morgan)
Date: Thursday, November 1, 2018
Time: 1:10pm to 2:00pm
Location: MUDD 303

View the abstract

Subhabrata Sen (MIT) | 11/8/2018 | 1:10pm to 2:00pm

Speaker: Subhabrata Sen (MIT)
Date: Thursday, November 8, 2018
Time: 1:10pm to 2:00pm
Location: MUDD 303

Title: Sampling convergence for random graphs: graphexes and multigraphexes

Abstract: We will look at structural properties of large, sparse random graphs through the lens of sampling convergence (Borgs, Chayes, Cohn and Veitch ’17). Sam- pling convergence generalizes left convergence to sparse graphs, and describes the limit in terms of a graphex. We will introduce this framework and motivate the components of a graphex. Subsequently, we will discuss the graphex limit for several sparse random (multi)graphs of practical interest.

This is based on joint work with Christian Borgs, Jennifer Chayes, and Souvik Dhara.

Bio: Subhabrata Sen is a Schramm Postdoctoral Fellow at the Department of Mathematics, Massachusetts Institute of Technology, and Microsoft Research (New England). He completed his Ph.D. in 2017 from the Department of Statis- tics, Stanford University, where he was advised jointly by Prof. Amir Dembo and Prof. Andrea Montanari. He was awarded the “Probability Dissertation Award” for his thesis “Optimization, Random Graphs, and Spin Glasses”. Sub- habrata’s research interests include random combinatorial optimization, random graphs, spin glasses, and hypothesis testing.

Mathieu Rosenbaum (Ecole Polytechnique) | 11/15/2018 | 1:10pm to 2:00pm

Speaker: Eunhye Song (Penn State)
Date: Thursday, November 15, 2018
Time: 1:10pm to 2:00pm
Location: MUDD 303

Speaker: Mathieu Rosenbaum (Ecole Polytechnique)

Title: No arbitrage implies power-law market impact and rough volatility

Abstract: Market impact is the link between the volume of a (large) order and the price move during and after the execution of this order. We show that under no-arbitrage assumption, the market impact function can only be of power-law type. Furthermore, we prove that this implies that the macroscopic price is diffusive with rough volatility, with a one-to-one correspondence between the exponent of the impact function and the Hurst parameter of the volatility. Hence we simply explain the universal rough behavior of the volatility as a consequence of the no-arbitrage property. From a mathematical viewpoint, our study relies in particular on new results about hyper-rough stochastic Volterra equations. This is joint work with Paul Jusselin.

Sebastian Engelke (U Geneva) | 11/29/2018 | 1:10pm to 2:00pm

Speaker: Sebastian Engelke (U Geneva)
Date: Thursday, November 29, 2018
Time: 1:10pm to 2:00pm
Location: MUDD 303

Title: Graphical Models and Structural Learning for Extremes

Abstract:

Conditional independence, graphical models and sparsity are key notions for parsimonious models in high dimensions and for learning structural relationships in the data. The theory of multivariate and spatial extremes describes the risk of rare events through asymptotically justified limit models such as max-stable and multivariate Pareto distributions. Statistical modeling in this field has been limited to moderate dimensions so far, owing to complicated likelihoods and a lack of understanding of the underlying probabilistic structures.

We introduce a general theory of conditional independence for multivariate Pareto distributions that allows to define graphical models and sparsity for extremes. New parametric models can be built in a modular way and statistical inference can be simplified to lower-dimensional margins. We define the extremal variogram, a new summary statistics that turns out to be a tree metric and therefore allows to efficiently learn an underlying tree structure through Prim's algorithm. For a popular parametric class of multivariate Pareto distributions we show that, similarly to the Gaussian case, the sparsity pattern of a general graphical model can be easily read of from suitable inverse covariance matrices. This enables the definition of an extremal graphical lasso that enforces sparsity in the dependence structure.

We illustrate the results with an application to flood risk assessment on the Danube river.

Bio:

Sebastian Engelke is assistant professor at the Research Center for Statistics at the University of Geneva. He is currently visiting at the Department of Statistical Sciences at the University of Toronto. Previously he was an Ambizione fellow at EPF Lausanne with Anthony Davison. Sebastian did his studies in Mathematics at University of Göttingen and UC Berkeley, and he finished his PhD as a Deutsche Telekom Foundation fellow in 2013 at the University of Göttingen with Martin Schlather. His research interests are in extreme value theory, spatial statistics, graphical models and data science.

Web: http://www.sengelke.com/

Eunhye Song (Penn State) | 12/6/2018 | 1:10pm to 2:00pm

Speaker: Eunhye Song (Penn State)
Date: Thursday, December 6, 2018
Time: 1:10pm to 2:00pm
Location: MUDD 303

Title: Solving a large-scale discrete simulation optimization problem using Gaussian Markov random fields

Abstract: Gaussian Process (GP)-based inferential optimization, sometimes referred to as Bayesian optimization, has gained its popularity for optimizing a function that is mildly expensive to evaluate and has no known structural properties. In this talk, we discuss solving a discrete simulation optimization problem with a combinatorially large solution space using the Gaussian Markov Improvement Algorithm (GMIA), a Gaussian Markov random field-based inferential optimization method. GMRF is a GP defined on a graph of solutions whose precision matrix's sparsity pattern is decided by the edges of the graph. At each iteration, GMIA updates the conditional distribution of the GMRF based on the simulated solutions and selects the next solution to simulate based on the complete expected improvement (CEI) criterion, which can be also used as a stopping criterion. GMIA is globally convergent to the optimal solution when the simulation budget increases to infinity and shows excellent empirical finite-sample performance as it 1) simulates only a small fraction of the solution space, and 2) stops correctly when the desired optimality gap is achieved. When the solution space is large, updating the GMRF and evaluating the CEIs for all feasible solutions can be computationally challenging. We introduce two modifications of GMIA to alleviate such challenges. The first is the restricted search set scheme, which periodically forms a set of promising solutions and restricts allocating simulation effort only to those solutions for p iterations (or until a stopping criterion is met). By doing so, we avoid factorizing a large precision matrix at each iteration while computing the CEIs of the solutions included in the search set exactly. The second is the multi-resolution GMIA (MR-GMIA) based on multiple layers of GMRFs defined on the feasible solution space. The solution space is divided into subregions, where each subregion is modeled by a solution-level GMRF and the subregions become “solutions” to the region-level GMRF. As a result, both solution-level and region-level GMRFs enjoy smaller graphs compared to the original solution space.

Bio: Eunhye Song is Harold and Inge Marcus Early Career Assistant Professor in Industrial and Manufacturing Engineering at Penn State University. She earned her PhD degree in Industrial Engineering and Management Sciences at Northwestern University in 2017 and MS and BS in Industrial and Systems Engineering at Korea Advanced Institute of Science and Technology (KAIST) in 2012 and 2010, respectively. Her research interests include design of simulation experiments, large-scale discrete simulation optimization, input uncertainty quantification, and simulation optimization in the presence of model risk. She collaborated with Simio, a leading discrete-event simulation software company, on developing a statistical tool to quantify input uncertainty for a Simio model, which is now a standard part of Simio's software product. She also worked with General Motors' R&D group on global sensitivity analysis of Vehicle Content Optimization simulator, which GM uses to find the optimal vehicle content portfolios of their major vehicle lines to maximize GM's market share and profit. She is an active member of INFORMS Simulation Society (I-Sim) and currently serving on the I-Sim Diversity Committee. itle: Solving a large-scale discrete simulation optimization problem using Gaussian Markov random fields

Abstract: Gaussian Process (GP)-based inferential optimization, sometimes referred to as Bayesian optimization, has gained its popularity for optimizing a function that is mildly expensive to evaluate and has no known structural properties. In this talk, we discuss solving a discrete simulation optimization problem with a combinatorially large solution space using the Gaussian Markov Improvement Algorithm (GMIA), a Gaussian Markov random field-based inferential optimization method. GMRF is a GP defined on a graph of solutions whose precision matrix's sparsity pattern is decided by the edges of the graph. At each iteration, GMIA updates the conditional distribution of the GMRF based on the simulated solutions and selects the next solution to simulate based on the complete expected improvement (CEI) criterion, which can be also used as a stopping criterion. GMIA is globally convergent to the optimal solution when the simulation budget increases to infinity and shows excellent empirical finite-sample performance as it 1) simulates only a small fraction of the solution space, and 2) stops correctly when the desired optimality gap is achieved. When the solution space is large, updating the GMRF and evaluating the CEIs for all feasible solutions can be computationally challenging. We introduce two modifications of GMIA to alleviate such challenges. The first is the restricted search set scheme, which periodically forms a set of promising solutions and restricts allocating simulation effort only to those solutions for p iterations (or until a stopping criterion is met). By doing so, we avoid factorizing a large precision matrix at each iteration while computing the CEIs of the solutions included in the search set exactly. The second is the multi-resolution GMIA (MR-GMIA) based on multiple layers of GMRFs defined on the feasible solution space. The solution space is divided into subregions, where each subregion is modeled by a solution-level GMRF and the subregions become “solutions” to the region-level GMRF. As a result, both solution-level and region-level GMRFs enjoy smaller graphs compared to the original solution space.