Trees and recursive partitions are most well-known in supervised learning for predictive tasks, such as regression and classification. Famous examples include CART, random forest, and boosting and their Bayesian cousins such as Bayesian CART and BART. A natural question is whether such successes can be replicated in the context of unsupervised problems and modeling unlabeled data. In this short course, I will first survey some classical Bayesian generative models and nonparametric priors based on trees and partitions, followed by several more recent examples of tree-based approaches for unsupervised learning and generative modeling, where the two primary objectives are to (i) learn the underlying nature of complex multivariate, possibly high-dimensional distributions based on unlabeled i.i.d. training data, and (ii) generate new data samples from the trained model. In these examples, the employment of trees and partitions leads to highly efficient, statistically rigorous inference algorithms that scale approximately linearly in the sample size and accommodate moderately high (e.g., hundreds) dimensions. Some examples from biomedical applications such as microbiome compositional analysis will be provided.

This tutorial will focus on a number of applications of Bayesian nonparametric modelling to substantive issues in environment, sport, health and society. These projects have all been undertaken in collaboration with organisations that want to use the answers for decision-making. The Australian Antarctic Division asked: how large a boundary should we put around an oil spill based on soil contamination? The Queensland Academy of Sport asked: can we predict injuries so our players can win well? The Australian Government called for support and strategies to manage covid. The Red Cross cares about understanding and predicting natiomal and global terrorism events. In answering these questions, we will discuss a variety of models and computational algorithms. We will also have hands-on activities and invite discussion about improvements and alternativies, implementation and translation. The research discussed in the tutorial was led by Julyan Arbel, Raiha Browning, John Worrall and Judith Rousseau.

The fundamental theorem of Bayesian statistics is a symmetry result: de Finetti's theorem characterizes distributions that are invariant under permutations. Invariance under a class of transformations is just how physicists and mathematicians define symmetry. In the last few years, symmetry is suddenly being discovered and rediscovered everywhere: In generative modeling, in data augmentation, in conformal prediction, in machine learning for science, and so forth. I will try to explain in the simplest possible terms that these ideas are all related mathematically; that exchangeability implies not just the de Finetti representation, but also the law of large numbers and the central limit theorem; that this is not only true for exchangeability, but for a much larger class of symmetries; and how all of this relates to work in machine learning that studies symmetries of functions rather than distributions. I will also sketch some open problems. There is a lot of work to be done.