Andres Felipe Barrientos
On MacEachern’s Dependent Dirichlet Process: Flexible Bayesian Predictor-Dependent Modeling
The Dependent Dirichlet Process (DDP) is a fundamental tool in Bayesian nonparametric (BNP) methods as it allows for flexible modeling of an unknown distribution that evolves with predictors. In this talk, we provide a brief overview of the DDP, including commonly used special cases such as the single-weights and single-atoms DDP. We examine properties of the DDP that have contributed to its widespread use in BNP modeling, particularly focusing on topological support when used as a prior distribution. Then, we discuss how the DDP has been successfully employed as a building block for density regression, including some examples. Finally, we cover recent developments surrounding the DDP and provide concluding remarks on the DDP’s significant impact within the Bayesian nonparametric community.
Keywords: Dependent Dirichlet Process (DDP); Bayesian Nonparametrics (BNP); Density Regression; Stick-Breaking representation; Mixture models
Jim Griffin
DDP modelling using normalised random measures
MacEachern’s Dependent Dirichlet Process represents a seminal advance in Bayesian nonparametric modelling by providing a structure for building a wide-range of priors. To understand the importance and influence of this work, I will review its historical context and my own attempts to build flexible DDP-type priors using Compound Random Measures and Normalised Latent Factor Measures.
Griffin, J. E. and Leisen, F. (2017): Compound Random Measures and their Use in Bayesian Non-parametrics, Journal of the Royal Statistical Society, Series B, 79, 525-545.
Beraha, M. and Griffin, J. E. (2023): Normalised latent measure factor models, Journal of the Royal Statistical Society, Series B, 85, 1247-1270.
Antonio Lijoi
The DDP and random measures-based approaches
The Dependent Dirichlet Process (DDP), introduced by MacEachern (1999, 2000), has inspired a rich line of research on dependent random probability measures. While most extensions of the original framework have been developed using stick-breaking representations, this talk focuses on an alternative approach based on dependent completely random measures. We provide a concise overview of key developments in this area, illustrating how they have been shaped by Steve’s pioneering ideas. Finally, we outline some open problems that may be worth of further exploration.
Ramses Mena
Continuous time Bayesian nonparametric dependent processes
Building on the concept of the dependent Dirichlet process (MacEachern 1999, 2000), we propose a general framework for constructing continuous-time-dependent nonparametric priors. We examine their key properties and discuss their implications for the statistical modeling of random phenomena evolving in continuous time. Finally, we demonstrate their practical impact through real-data applications.
Joint work with: Luis Gutierrez, Matteo Ruggiero, Stephen Walker
Keywords: Continuous time; Diffusion processes; Markov processes; Mixture models
David Dunson
Logistic-beta processes for dependent random probabilities with beta marginals
The beta distribution serves as a canonical tool for modeling probabilities in statistics and machine learning. However, there is limited work on flexible and computationally convenient stochastic process extensions for modeling dependent random probabilities. We propose a novel stochastic process called the logistic-beta process, whose logistic transformation yields a stochastic process with common beta marginals. Logistic-beta processes can model dependence on both discrete and continuous domains, such as space or time, and have a flexible dependence structure through correlation kernels. Moreover, its normal variance-mean mixture representation leads to effective posterior inference algorithms. We show how the proposed logistic-beta process can be used to design computationally tractable dependent Bayesian nonparametric models, including dependent Dirichlet processes and extensions. We illustrate the benefits through nonparametric binary regression and conditional density estimation examples, both in simulation studies and in a pregnancy outcome application.
Joint work with: Changwoo J. Lee, Alessandro Zito, Huiyan Sang.
Tamara Broderick
The dependent Dirichlet process and local exchangeability
Bayesian models that make exchangeable cluster assignments according to cluster frequencies drawn with a Dirichlet process are ubiquitous. In many practical settings, it is more realistic that the cluster frequencies (and cluster identities) can change over covariates such as time. The dependent Dirichlet process (DDP) offers practical modeling and inference in this setting when additional regularity assumptions hold—such as smoothness of the process as a function of the covariates. In our own work, we view such a DDP as a particular, practical instance of what we call “local exchangeability”—where swapping data associated with nearby covariates causes a bounded change in the distribution. We prove that locally exchangeable processes correspond to independent observations from an underlying measure-valued stochastic process. Using this main probabilistic result, we show that the local empirical measure of a finite collection of observations provides an approximation of the underlying measure-valued process and Bayesian posterior predictive distributions.