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William Weimin Yoo

Publications and source records attributed to William Weimin Yoo.

10 recordsLinked to original sources
arXiv2019

Bayesian cluster analysis: Point estimation and credible balls

Clustering is widely studied in statistics and machine learning, with applications in a variety of fields. As opposed to classical algorithms which return a single clustering solution, Bayesian nonparametric models provide a posterior over the entire space of partitions, allowing one to assess statistical properties, such as uncertainty on the number of clusters. However, an important problem is how to summarize the posterior; the huge dimension of partition space and difficulties in visualizing it add to this problem. In a Bayesian analysis, the posterior of a real-valued parameter of interest is often summarized by reporting a point estimate such as the posterior mean along with 95% credible intervals to characterize uncertainty. In this paper, we extend these ideas to develop appropriate point estimates and credible sets to summarize the posterior of clustering structure based on decision and information theoretic techniques.

stat.ME↗
arXiv2018

SciSports: Learning football kinematics through two-dimensional tracking data

SciSports is a Dutch startup company specializing in football analytics. This paper describes a joint research effort with SciSports, during the Study Group Mathematics with Industry 2018 at Eindhoven, the Netherlands. The main challenge that we addressed was to automatically process empirical football players' trajectories, in order to extract useful information from them. The data provided to us was two-dimensional positional data during entire matches. We developed methods based on Newtonian mechanics and the Kalman filter, Generative Adversarial Nets and Variational Autoencoders. In addition, we trained a discriminator network to recognize and discern different movement patterns of players. The Kalman-filter approach yields an interpretable model, in which a small number of player-dependent parameters can be fit; in theory this could be used to distinguish among players. The Generative-Adversarial-Nets approach appears promising in theory, and some initial tests showed an improvement with respect to the baseline, but the limits in time and computational power meant that we could not fully explore it. We also trained a Discriminator network to distinguish between two players based on their trajectories; after training, the network managed to distinguish between some pairs of players, but not between others. After training, the Variational Autoencoders generated trajectories that are difficult to distinguish, visually, from the data. These experiments provide an indication that deep generative models can learn the underlying structure and statistics of football players' trajectories. This can serve as a starting point for determining player qualities based on such trajectory data.

cs.LG↗
arXiv2018

Adaptive Supremum Norm Posterior Contraction: Wavelet Spike-and-Slab and Anisotropic Besov Spaces

Supremum norm loss is intuitively more meaningful to quantify function estimation error in statistics. In the context of multivariate nonparametric regression with unknown error, we propose a Bayesian procedure based on spike-and-slab prior and wavelet projections to estimate the regression function and all its mixed partial derivatives. We show that their posterior distributions contract to the truth optimally and adaptively under supremum-norm loss. The master theorem through tests with exponential errors used in Bayesian nonparametrics was not adequate to deal with this problem, and we developed a new idea such that posterior under the regression model is systematically reduced to a posterior arising from some quasi-white noise model, where the latter model greatly simplifies our rate calculations. Hence, this paper takes the first step in showing explicitly how one can translate results from white noise to regression model in a Bayesian setting.

math.ST↗
arXiv2018

Discussion on Using Stacking to Average Bayesian Predictive Distributions by Yao et al

I begin by summarizing key ideas of the paper under discussion. Then I will talk about a graphical modeling perspective, posterior contraction rates and alternative methods of aggregation. Moreover, I will also discuss possible applications of the stacking method to other problems, in particular, aggregating (sub)posterior distributions in distributed computing.

stat.OT↗
arXiv2018

Bayesian mode and maximum estimation and accelerated rates of contraction

We study the problem of estimating the mode and maximum of an unknown regression function in the presence of noise. We adopt the Bayesian approach by using tensor-product B-splines and endowing the coefficients with Gaussian priors. In the usual fixed-in-advanced sampling plan, we establish posterior contraction rates for mode and maximum and show that they coincide with the minimax rates for this problem. To quantify estimation uncertainty, we construct credible sets for these two quantities that have high coverage probabilities with optimal sizes. If one is allowed to collect data sequentially, we further propose a Bayesian two-stage estimation procedure, where a second stage posterior is built based on samples collected within a credible set constructed from a first stage posterior. Under appropriate conditions on the radius of this credible set, we can accelerate optimal contraction rates from the fixed-in-advanced setting to the minimax sequential rates. A simulation experiment shows that our Bayesian two-stage procedure outperforms single-stage procedure and also slightly improves upon a non-Bayesian two-stage procedure.

math.ST↗
arXiv2018

Discussion on Computationally Efficient Multivariate Spatio-Temporal Models for High-Dimensional Count-Valued Data by Bradley et al

I begin my discussion by summarizing the methodology proposed and new distributional results on multivariate log-Gamma derived in the paper. Then, I draw an interesting connection between their work with mean field variational Bayes. Lastly, I make some comments on the simulation results and the performance of the proposed Poisson multivariate spatio-temporal mixed effects model (P-MSTM).

stat.ME↗
arXiv2017

The Bayes Lepski's Method and Credible Bands through Volume of Tubular Neighborhoods

For a general class of priors based on random series basis expansion, we develop the Bayes Lepski's method to estimate unknown regression function. In this approach, the series truncation point is determined based on a stopping rule that balances the posterior mean bias and the posterior standard deviation. Equipped with this mechanism, we present a method to construct adaptive Bayesian credible bands, where this statistical task is reformulated into a problem in geometry, and the band's radius is computed based on finding the volume of certain tubular neighborhood embedded on a unit sphere. We consider two special cases involving B-splines and wavelets, and discuss some interesting consequences such as the uncertainty principle and self-similarity. Lastly, we show how to program the Bayes Lepski stopping rule on a computer, and numerical simulations in conjunction with our theoretical investigations concur that this is a promising Bayesian uncertainty quantification procedure.

math.ST↗
arXiv2015

Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression

In the setting of nonparametric multivariate regression with unknown error variance, we study asymptotic properties of a Bayesian method for estimating a regression function f and its mixed partial derivatives. We use a random series of tensor product of B-splines with normal basis coefficients as a prior for f, and the error variance is either estimated using the empirical Bayes approach or is endowed with a suitable prior in a hierarchical Bayes approach. We establish pointwise, L2 and supremum norm posterior contraction rates for f and its mixed partial derivatives, and show that they coincide with the minimax rates. Our results cover even the anisotropic situation, where the true regression function may have different smoothness in different directions. Using the convergence bounds, we show that pointwise, L2 and supremum norm credible sets for f and its mixed partial derivatives have guaranteed frequentist coverage with optimal size. New results on tensor products of B-splines are also obtained in the course.

math.ST↗