Colloquium

Title: Blessing of dimension in Bayesian inference on covariance matrices

Abstract: Bayesian factor analysis is routinely used for dimensionality reduction in modeling of high-dimensional covariance matrices. Factor analytic decompositions express the covariance as a sum of a low rank and diagonal matrix. In practice, Gibbs sampling algorithms are typically used for posterior computation, alternating between updating the latent factors, loadings, and residual variances. In this article, we exploit a blessing of dimensionality to develop a provably accurate posterior approximation for the covariance matrix that bypasses the need for Gibbs or other variants of Markov chain Monte Carlo sampling. Our proposed Factor Analysis with BLEssing of dimensionality (FABLE) approach relies on a first-stage singular value decomposition (SVD) to estimate the latent factors, and then defines a jointly conjugate prior for the loadings and residual variances. The accuracy of the resulting posterior approximation for the covariance improves with increasing samples as well as increasing dimensionality. We show that FABLE has excellent performance in high-dimensional covariance matrix estimation, including producing well-calibrated credible intervals, both theoretically and through simulation experiments. We also demonstrate the strength of our approach in terms of accurate inference and computational efficiency by applying it to a gene expression dataset.

Title: From Divination to Gambling to Annuities

Abstract: This talk traces the history of the development of annuities,
starting from humans' earliest efforts at divination, proceeding to the
study of gambling, and thence to annuities.  Along the way, we will see
how divination has occurred on every populated continent, and similarly
for gambling.  Finally, we will describe how annuities were developed by
some of the earliest practitioners of mathematical probability.

Title: Outlier-robust posterior inference for linear models (Kaoru Irie, Associate Professor, Faculty of Economics, the University of Tokyo)

Abstract: Robust statistics, or outlier-robust inference, has been a focus of statistical research for many years and has produced abundant research results. In the Bayesian framework, however, research on outlier-robust "posterior" inference has advanced dramatically in the last 10 years. The key concept in modeling outlier-contaminated observations is the use of a super-heavy-tailed distribution, such as the log-Pareto distribution, as the error distribution. It has also been shown across multiple settings that the Student's t-distribution is not sufficiently heavy-tailed to accommodate outliers without affecting the posterior distribution.

In this talk, we review the recent literature on so-called posterior robustness, then showcase a series of our research, focusing on three classes of models: hierarchical linear models, Poisson/negative binomial linear models, and multivariate extensions. First, we construct a new super-heavy-tailed error distribution for hierarchical linear models using the variance mixture of normals, which is conditionally conjugate and allows a Gibbs sampler, while theoretically guaranteeing robustness of the posterior distribution.

Second, we consider the generalized linear models, specifically the Poisson/negative binomial sampling distributions for count-valued responses. In this research, we treat inflated zeros and small counts as outliers, alongside extremely large counts, within a robust statistics framework. Third, we extend the use of super-heavy-tailed distributions to multivariate linear models, including graphical models. To protect the correlation parameter against the effects of outliers, we propose introducing a real-valued latent variable and multiplying it by the standard deviation parameter based on the variance-correlation decomposition. Other covariance models are shown to lack posterior robustness or to require stronger assumptions for robustness.

This is joint work with Yasuyuki Hamura (Kyoto U) and Shonosuke Sugasawa (Keio U).

Title: Scalable Bayesian Spatial Modeling and Agent-Based Simulation for Ecological Decision Support

Abstract: Ecological management problems often require linking uncertain data to scenario-based predictions about how human activity may affect wildlife. This is challenging when observations are sparse, collected across multiple observation windows, and the quantities needed for decisions are not direct outputs of standard statistical models. In this talk I will describe a modeling framework that connects Bayesian spatial inference with simulation to support decision-making under uncertainty

The first component is a computationally efficient, recursive Bayesian approach for fitting spatial point process models to point pattern data derived from long-term monitoring and aerial imagery with non-overlapping sampling frames. This provides flexible estimates of spatial structure and yields posterior predictive spatial fields that can be carried forward into downstream analyses. The second component uses these posterior predictions to initialize a custom agent-based model that represents interactions among environmental conditions, human activity, and animal responses, with outputs summarized through disturbance metrics. The agent-based model is built as a modular system so that alternative submodels can be incorporated, including demographic components such as matrix population models and decision-oriented components such as Markov decision processes.