Data driven discovery is the modern trend of science. A plethora of developed models are dedicated in analyzing or assimilating data arising from problems in material science and chemistry to national defense and health. The proposed mini-symposium will focus on the uncertainty of data, and its speakers will discuss techniques of uncertainty quantification, parameter estimation and noise in complex data so that robust, reproducible and convergent results are propagated. By the same token, audience and speakers will benefit from a dynamic set of prominent and auspicious speakers with heterogeneous backgrounds spanning almost the entire spectrum of mathematical sciences, from topology and geometry to statistics and machine learning.
08:30
On the choice of importance distributions for multiple importance sampling estimators
Evangelos Evangelou | University of Bath | United Kingdom
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Evangelos Evangelou | University of Bath | United Kingdom
Importance sampling (IS) is a popular Monte Carlo procedure where samples from one distribution are weighted to estimate means with respect to others. The naive IS estimator, based on a single importance density, suffers from high variance if it is not `close' to the target density. There are situations where simultaneous estimation of means with respect to a large set of pdfs arise. The problems of the naive IS estimator are exacerbated in this case, as a single (importance) density may not work for all target pdfs. In this talk, the issues with naive IS will be demonstrated when used to evaluate the partition function of the Ising model. To reduce the variability of the naive IS estimator, one may use samples from multiple densities, giving rise to the so-called multiple IS estimator. I will discuss one such estimator based on reweighting samples from a mixture distribution and present a central limit theorem for the asymptotic distribution of this estimator. Finally, I will present clever ways of choosing these importance densities to achieve low variability.
09:00
Geometric and Topological Data Analysis of Enzyme Kinetics
Lewis Marsh | Oxford University | United Kingdom
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Lewis Marsh | Oxford University | United Kingdom
In this talk, we will mathematically study a differential equation model and generated data describing enzyme kinetics. With tools from computational algebra, geometry and topology, we perform a model reduction, study structural identifiability and infer parameter distributions to then quantify the shape of these. We use these results to show that various genetic perturbations have a different effect on enzyme kinetics (in parameter space), which seems to be interpretable in the biological system.
09:30
A Bayesian Framework for Persistent Homology
Vasileios Maroulas | University of Tennessee | United States
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Vasileios Maroulas | University of Tennessee | United States
Persistence diagrams offer a way to summarize topological and geometric properties latent in datasets. While several methods have been developed that utilize persistence diagrams in statistical inference, a full Bayesian treatment remains absent. This talk, relying on the theory of point processes, presents a Bayesian framework for inference with persistence diagrams relying on a substitution likelihood argument. In essence, we model persistence diagrams as Poisson point processes with prior intensities and compute posterior intensities by adopting techniques from the theory of marked point processes. We then propose a family of conjugate prior intensities via Gaussian mixtures to obtain a closed form of the posterior intensity. Finally we demonstrate the utility of this Bayesian framework, packaged in R under BayesTDA, with a classification problem using Bayes factors.
10:00
Bayesian Inference using the Pullback of a Persistence Map
Christopher Oballe | University of Tennessee | United States
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Christopher Oballe | University of Tennessee | United States
Recently, paradigms for Bayesian inference with topological data analysis have been introduced to quantify uncertainty in signals; these rely largely on point process models and substitution likelihood arguments. For the first time, we introduce a Bayesian framework that considers the pullback of topological summaries to the underlying data in its definition of likelihood. Our approach unifies topological data analysis and Bayesian filtering for time series. After building our framework, applications with neuroimaging data are presented.