An evolving probability for a binary event is naturally modeled as a bounded (so-called “win-”) martingale. Motivated by prediction markets and recent work on Aldous’ "most exciting game," we consider two questions. Can a given win-martingale be realized as the exact Bayesian posterior of a sequential experiment? And, if observing the process is costly, when should a decision maker stop and make a prediction?

We answer both questions for a broad class of martingale diffusions with separable volatility. We construct an explicit binary diffusion experiment whose posterior has the prescribed dynamics and show that the optimal strategy for a large class of sequential decision problems is to stop and predict according to two time-dependent thresholds. For this class of problems, we establish regularity and a unique integral equation characterization of the resulting stopping boundaries, and illustrate how the rate of information arrival can produce shrinking, expanding, or nonmonotone continuation regions. This is joint work with Karl Kristian Engelund (University of Copenhagen.)

Variational inference (VI) is a popular alternative to Markov chain Monte Carlo (MCMC) for approximating high-dimensional target distributions. At its core, VI approximates a high-dimensional target distribution—typically specified via an unnormalized density—by a simpler variational family. Despite its empirical success, the theoretical properties of variational inference have only recently begun to be understood. In the first part of the talk, I will discuss the robustness of mean-field variational inference (MFVI) through the lens of optimal transport. We aim to answer the following question: How sensitive are variational solutions to small perturbations of the target measure arising from changes in the underlying statistical model, likelihood, or prior? The main message is that stability depends sharply on the geometry of the target distribution. When the target is strongly log-concave, MFVI is quantitatively stable under perturbations of the target measure, yielding robustness guarantees for approximate Bayesian inference. In contrast, even simple non-log-concave targets, such as mixtures of two Gaussians, can cause MFVI to exhibit an instability known as mode collapse, revealing a fundamental lack of robustness to multimodality.

In the second part of the talk, I will present our work on the statistical and computational theory of a class of structured variational inference methods in which the variational family consists of all star-shaped distributions. We establish quantitative approximation guarantees and provide a polynomial-time algorithm for solving the VI problem when the target distribution is strongly log-concave. We also discuss concrete examples, including generalized linear models with Gaussian likelihoods. This talk is based on joint work with Bohan Wu, Alberto González-Sanz, Marcel Nutz, Sinho Chewi, Binghe Zhu, and Aram Pooladian.

Turnover, or change in the composition of species over space and time, is one of the primary ways to define beta diversity. Inferring what factors impact beta diversity is important not only for understanding biodiversity processes but also for conservation planning. A popular approach to understanding the drivers of compositional turnover is generalized dissimilarity modeling (GDM). However, the current GDM approach has several limitations. We provide an alternative approach that remedies these issues. In particular, we propose a model with a flexible mean function, complex spatial and spatio-temporal random effects that capture dependence unaccounted for by explanatory variables, and heterogeneous variance structure. Further, these features are offered in a model that can handle a large incidence of total dissimilarity through “1-inflation.” Such inflation would be expected in highly biodiverse areas with steep turnover gradients in space and time.

The models are implemented in a Bayesian framework, employing hierarchical specifications to yield full regression and spatial predictive inference, both with associated uncertainties. We illustrate the approach by examining dissimilarity in plant abundance surveys from the Greater Cape Floristic Region (GCFR) of South Africa. We model both at the species scale and the family scale, selecting the best model using out-of-sample predictive performance. Notably, the spatial random effects play a more important role in the modeling than all of the environmental variables. (This is joint work with Phil White, Henry Frye, Jasper Slingsby, and John Silander.)