{"id":38343,"date":"2023-02-05T23:02:33","date_gmt":"2023-02-06T04:02:33","guid":{"rendered":"https:\/\/www.bu.edu\/cise\/?p=38343"},"modified":"2023-04-24T11:48:59","modified_gmt":"2023-04-24T15:48:59","slug":"zachary-bezemek-probability-rare-events","status":"publish","type":"post","link":"https:\/\/www.bu.edu\/cise\/zachary-bezemek-probability-rare-events\/","title":{"rendered":"Estimating the Improbable with PhD Student Zack Bezemek"},"content":{"rendered":"

Could oil price volatility cause a stock market plunge? What\u2019s the risk factor of a new drug causing a seizure? How likely is it that a driverless car could crash into a bus?<\/p>\n

What\u2019s common among these seemingly disparate topics is that they are \u201crare events\u201d –\u00a0 <\/span>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>, a fifth-year PhD candidate of Mathematics, advised by Professor Konstantinos Spiliopoulos,<\/a> (Math\/Stats), a faculty affiliate of the Hariri Institute of Computing and the Center for Information and Systems Engineering.<\/p>\n

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Bezemek\u2019s research interests lie in the areas of stochastic processes, probability theory, and applied mathematics. He is interested in exploring dynamical properties of such models, as well as the connections between different equations which can result from constructing them within different mathematical frameworks or by taking different asymptotic limits.<\/em><\/figcaption><\/figure>\n

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.\u00a0 <\/span>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.\u00a0 <\/span><\/p>\n

Estimating probability requires simulating a target event multiple times. For infrequent events, lack of data is a problem.\u00a0Since 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.<\/p>\n

Bezemek\u2019s work aims to estimate the probability of rare events for complex IPS in analytically simpler and computationally efficient ways.<\/p>\n

In his paper \u201cLarge deviations for interacting multiscale particle systems<\/a><\/span>,\u201d he not only estimates the probability of rare events for IPS but also the impact of the system\u2019s 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\u00a0\u201cthe future is independent of the past given the present.\u201d<\/span>\u00a0Large deviations analysis provides a \u201crate function,\u201d which characterizes the decay rates of rare event probabilities.<\/p>\n

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\u00a0provides 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.<\/span><\/p>\n

\u201cZack 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,\u201d says Spiliopoulos. \u201cThis 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\u2019s proof.\u201d<\/p>\n

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”,<\/a><\/span> they provide provably-efficient importance sampling Monte Carlo schemes for the estimation of these complex event types.\u00a0<\/span><\/p>\n

\u201cThis is the first paper rigorously addressing this problem and it is expected to be a very influential result,\u201d says Spiliopoulos.<\/p>\n

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.\u00a0 <\/span>Such systems appear in a plethora of applications ranging from social and opinion dynamics to molecule interaction and disease spreading models.<\/p>\n

Bezemek first got interested in modeling stochastic processes during \u00a0a summer undergraduate research opportunity at University of Michigan-Dearborn. \u00a0<\/span>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.<\/p>\n

\u201cBeing 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,\u201d says Bezemek. \u201cIt 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.\u201d<\/p>\n

Bezemek\u2019s research could be applied to help break barriers between different fields of science and facilitate collaboration and communication. \u201cMy work can be considered as a stepping stone towards better understanding various phenomena in diverse fields,\u201d says Bezemek. \u201cIn 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.\u201d<\/p>\n

\u00a0<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Could oil price volatility cause a stock market plunge? What\u2019s the risk factor of a new drug causing a seizure? How likely is it that a driverless car could crash into a bus? What\u2019s common among these seemingly disparate topics is that they are \u201crare events\u201d –\u00a0 low-probability incidents that, if they occur, often have […]<\/p>\n","protected":false},"author":10316,"featured_media":38352,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[76,205],"tags":[],"_links":{"self":[{"href":"https:\/\/www.bu.edu\/cise\/wp-json\/wp\/v2\/posts\/38343"}],"collection":[{"href":"https:\/\/www.bu.edu\/cise\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.bu.edu\/cise\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.bu.edu\/cise\/wp-json\/wp\/v2\/users\/10316"}],"replies":[{"embeddable":true,"href":"https:\/\/www.bu.edu\/cise\/wp-json\/wp\/v2\/comments?post=38343"}],"version-history":[{"count":20,"href":"https:\/\/www.bu.edu\/cise\/wp-json\/wp\/v2\/posts\/38343\/revisions"}],"predecessor-version":[{"id":38733,"href":"https:\/\/www.bu.edu\/cise\/wp-json\/wp\/v2\/posts\/38343\/revisions\/38733"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.bu.edu\/cise\/wp-json\/wp\/v2\/media\/38352"}],"wp:attachment":[{"href":"https:\/\/www.bu.edu\/cise\/wp-json\/wp\/v2\/media?parent=38343"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.bu.edu\/cise\/wp-json\/wp\/v2\/categories?post=38343"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.bu.edu\/cise\/wp-json\/wp\/v2\/tags?post=38343"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}