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# Archive: Probability

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 Robert Webber (Courant Inst., NYU) Robert Webber (Courant Inst., NYU) Zoom
Title: A splitting method to reduce MCMC variance
Abstract: We explore whether splitting and killing methods can improve the accuracy of Markov chain Monte Carlo (MCMC) estimates of rare event probabilities, and we make three contributions. First, we prove that "weighted ensemble" is the only splitting and killing method that provides asymptotically consistent estimates when combined with MCMC. Second, we prove a lower bound on the asymptotic variance of weighted ensemble's estimates. Third, we give a constructive proof and numerical examples to show that weighted ensemble can approach this optimal variance bound, in many cases reducing the variance of MCMC estimates by multiple orders of magnitude. 3/24/2021 5:00:00 PM 3/24/2021 6:00:00 PM False Ross Pinsky (Technion) Ross Pinsky (Technion) EWG 336 Title: Natural Probabilistic Model on the Integers and its Relation to Dickman-Type Distributions and Buchstab’s Function

Abstract: Click here to view the abstract 2/17/2020 6:30:00 PM 2/17/2020 7:30:00 PM False Kelvin Rivera-Lopez (UDel) Kelvin Rivera-Lopez (UDel) EWG 336 Title: An Introduction to One-Dimensional Markov Processes: Part III

Abstract: In this talk, we'll demonstrate how generators can be used to uncover some interesting properties of Feller semigroups. One such property is that Feller semigroups can be suitably approximated by semigroups of pseudo-Poisson processes. If time permits, we'll discuss the generator associated with Brownian Motion. 10/7/2019 5:30:00 PM 10/7/2019 6:30:00 PM False Markos Katsoulakis, UMASS Markos Katsoulakis, UMASS EWG336 Title: Information Theory, Robust Uncertainty Quantification and Predictive Guarantees.

Abstract: We discuss connections between information theory, statistical learning, uncertainty quantification and predictive modeling, and in particular how to systematically select probabilistic metrics for enhanced learning and prediction. In machine learning, uncertainty quantification, as well as in model selection, reduction and approximate inference, we typically use a variety of probability metrics and information divergences, e.g. Wasserstein, Kullback-Leibler (KL), Renyi, χ^2 or Hellinger metrics. Although some choices are natural e.g. the relation between the KL divergence and Maximum Likelihood, often selecting probability metrics may appear arbitrary or becomes justified only a posteriori, based on the success of our final goal. To address these questions we focus instead on the impact of probability metrics on the tasks at hand, e.g. on predicting given observables or carrying out designated statistical learning tasks such as coarse-graining. To this end, this perspective requires to relate probability metrics/divergences with observables. Here we discuss some recently derived information inequalities that clarify and classify the connection between metrics and tasks to be performed; for example, the KL-divergence (the average of the log-likelihood between probabilities) leads to tight and computable information inequalities to control typical" observables, e.g. expected values and variances. The family of Renyi divergences (related to the cumulant generating function of the log-likelihood) allows for information inequalities for rare events and related risk-sensitive observables. Finally in the context of sensitivity analysis the Fisher Information (the covariance of the score function) controls sensitivities of expected values, while the cumulant generating function of the score controls the sensitivity of rare events. All these metrics can be used in conjunction with concentration inequalities for easier implementation or to account for finite data. Finally we demonstrate these methods in complex, high dimensional reaction networks, and graphical modeling for multiscale modeling of energy storage devices. 9/30/2019 5:30:00 PM 9/30/2019 6:30:00 PM False