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Bayesian nonparametric statistics, St-Flour lecture notes

Ismaël Castillo

Abstract

These are lecture notes of the 51st Saint-Flour summer school, July 2023, on the topic of Bayesian nonparametric statistics

Bayesian nonparametric statistics, St-Flour lecture notes

Abstract

These are lecture notes of the 51st Saint-Flour summer school, July 2023, on the topic of Bayesian nonparametric statistics
Paper Structure (89 sections, 5 theorems, 718 equations, 12 figures)

This paper contains 89 sections, 5 theorems, 718 equations, 12 figures.

Table of Contents

  1. Introduction, rates I
  2. Statistical models
  3. Nonparametric models
  4. High-dimensional models
  5. Semiparametric models
  6. The Bayesian paradigm
  7. Nonparametric priors, examples
  8. Convergence of the posterior distribution
  9. A generic result, first version
  10. Testing and entropy, a second generic result
  11. An example for which explicit computations are possible
  12. First step: reduction to a mean/variance problem.
  13. Study of the terms (a) and (b).
  14. Rates II and first examples
  15. Extensions
  16. ...and 74 more sections

Key Result

Corollary 1

Let $W$ be a mean-zero Gaussian process taken as prior on $f$ in Gaussian white noise. Let $\psi(f) = \int_0^1 f a$ be a linear functional and consider the two cases: Then the posterior distribution of $\sqrt{n}(\psi(\eta) - \hat{\psi})$ converges weakly in $P_0-$probability to a Gaussian distribution with mean 0 and variance $\|a\|_2^2$ if

Figures (12)

  • Figure 2.1: Indexed binary tree with levels $l\le 2$ represented. The nodes index the intervals $I_\varepsilon$. Edges are labelled with random variables $Y_\varepsilon$.
  • Figure 3.1: Finite tree of activated coefficients
  • Figure 4.1: Structure of neural network with one hidden layer
  • Figure 4.2: Operations at the level of the first neuron and output
  • Figure 4.3: Neural network with $2$ hidden layers
  • ...and 7 more figures

Theorems & Definitions (5)

  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Corollary 5