This up-to-date overview of Bayesian nonparametric statistics provides both an introduction to the field and coverage of recent research topics, including deep neural networks, high-dimensional models and multiple testing, Bernstein-von Mises theorems and variational Bayes approximations, many of which have previously only been accessible through research articles. Although Bayesian posterior distributions are widely applied in astrophysics, inverse problems, genomics, machine learning and elsewhere, their theory is still only partially understood, especially in complex settings such as nonparametric or semiparametric models. Here, the available theory on the frequentist analysis of posterior distributions is outlined in terms of convergence rates, limiting shape results and uncertainty quantification. Based on lecture notes for a course given at the St-Flour summer school in 2023, the book is aimed at researchers and graduate students in statistics and probability.

Gives an accessible up-to-date introduction to Bayesian nonparametric statistics Provides complete mathematical proofs or describes the key ideas Discusses recent topics such as deep neural networks and variational Bayes approximations

Autorentext

Ismaël Castillo studied mathematics at the École Normale Supérieure de Lyon and obtained a PhD in statistics from the Université Paris-Sud at Orsay in 2006. After a postdoc at the Vrije Universiteit in Amsterdam, in 2009 he became CNRS researcher in Paris, France. Since 2015 he has been full professor of Statistics at Sorbonne Université in Paris. He has taught statistics courses worldwide, especially in Bayesian inference, including invited lectures at Cambridge, Columbia, Berlin, Lunteren and St-Flour. He is an IMS fellow and an honorary fellow of the Institut Universitaire de France.

Inhalt

-1. Introduction, rates I.-2. Rates II and first examples.-3. Adaptation I: smoothness.-4. Adaptation II: high-dimensions and deep neural networks.- 5. Bernstein-von Mises I: functionals.- 6. Bernstein-von Mises II: multiscale and applications.- 7. classification and multiple testing.- 8. Variational approximations.