# Estimating the Improbable with PhD Student Zack Bezemek

Could oil price volatility cause a stock market plunge? What’s the risk factor of a new drug causing a seizure? How likely is it that a driverless car could crash into a bus?

What’s common among these seemingly disparate topics is that they are “rare events” – low-probability incidents that, if they occur, often have a high and devastating impact. Calculating the probability of such events is difficult and the focus of Zachary Bezemek, a fifth-year PhD candidate of Mathematics, advised by Professor Konstantinos Spiliopoulos, (Math/Stats), a faculty affiliate of the Hariri Institute of Computing and the Center for Information and Systems Engineering.

Bezemek estimates rare event probabilities for large Interacting Particle Systems (IPS). These are a type of probabilistic model that describes the collective behavior of randomly interacting components. These systems are a growing area of focus in interdisciplinary research to solve complex problems in natural, social, and human-made systems. Applications range from neuroscience (spiking neurons) and biological systems (genome instability and mutation) to engineering (autonomous traffic flow on highways), mathematical finance (stock market impacts), chemistry (free energy computation), opinion dynamics, and other fields.

Estimating probability requires simulating a target event multiple times. For infrequent events, lack of data is a problem. Since the event is rare, it might take repeating simulations on orders of magnitude in the millions or even billions to produce one rare event, never mind simulating it with enough frequency to create a probability estimation.

Bezemek’s work aims to estimate the probability of rare events for complex IPS in analytically simpler and computationally efficient ways.

In his paper “Large deviations for interacting multiscale particle systems,” he not only estimates the probability of rare events for IPS but also the impact of the system’s dynamical variables across different time scales. He studies agents or particles interacting with each other (influenced by the average behavior of the system) where different dynamical components of the systems can evolve on different time scales (multiscale structures). These slow-fast systems are driven by continuous-time, random processes and obey the Markov assumption that “the future is independent of the past given the present.” Large deviations analysis provides a “rate function,” which characterizes the decay rates of rare event probabilities.

According to Bezemek, the intuition behind this problem is to characterize rare events (large deviations) in the combined regime of infinite agents (or particles) and fast oscillations, meaning that the number of particles go to infinity as the magnitude of the time-scale separation parameter goes to zero. He provides a tractable macroscopic model for describing systems which have both features of multiscale structure and particle interactions. He also quantifies how well the tractable model approximates the more complicated microscopic model.

“Zack was not only able to derive the large deviations under the effect of multiple scales in the combined limit, but he was also able to rigorously show, in a very ingenious way, the equivalence of the two different formulations in both the absence and presence of multiple scales,” says Spiliopoulos. “This is a very impressive result for a fresh PhD student because it is rather technical and deep. The equivalence of the different forms of the rate functions was hypothesized by many authors but without any proof until Zack’s proof.”

On the computational level of these rare event probabilities, Bezemek began collaborating with Max Heldman (BU PhD student, Mathematics, at the time) to develop a computational scheme to accelerate the speed of computing probabilities of rare events associated with interacting particle systems. In their paper “Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions”, they provide provably-efficient importance sampling Monte Carlo schemes for the estimation of these complex event types.

“This is the first paper rigorously addressing this problem and it is expected to be a very influential result,” says Spiliopoulos.

Leveraging upon the machinery developed in this work will allow researchers to provably and efficiently compute rare event probabilities associated with a collection of agents or particles that could, for example, be acting like a group or be subject to repulsion and attraction forces. Such systems appear in a plethora of applications ranging from social and opinion dynamics to molecule interaction and disease spreading models.

Bezemek first got interested in modeling stochastic processes during a summer undergraduate research opportunity at University of Michigan-Dearborn. Working with Professor Hyejin Kim, he was looking at populations of cells that interact and sought to approximate the likelihood of cancer cells regrowing after they had been treated to a point where it was expected that the body could naturally eradicate the remaining population of cells.

“Being an undergraduate at the time, I lacked the theoretical background to prove a large deviations principle for this event, and the best I could do was crude numerical experiments that provided evidence that the large deviations principle should hold,” says Bezemek. “It was exciting to see the impact of mathematical probability on applications of consequence. The experience motivated me to pursue my doctoral studies in probability research.”

Bezemek’s research could be applied to help break barriers between different fields of science and facilitate collaboration and communication. “My work can be considered as a stepping stone towards better understanding various phenomena in diverse fields,” says Bezemek. “In the future, a major goal for me is to be able to engage in interdisciplinary research and work with researchers in biology, chemistry, or artificial intelligence to see how this theory can be applied to answer questions relevant to their fields.”