Summary of [Ferguson 1973] (A Bayesian analysis of some nonparametric problems)

1. A Bayesian Analysis of Some
Nonparametric Problems
(Ferguson, 1973)
2016.07.04
Sangwoo Mo

2. Table of Contents
• Brief Introduction
• Dirichlet Process
• Applications

3. Table of Contents
• Brief Introduction
• Dirichlet Process
• Applications

4. Brief Introduction
• In Bayesian Nonparametrics, we need a prior such that
1. support of prior is large
2. posterior is manageable
• (Ferguson, 1973) is the first paper proposed Dirichlet
Process that satisfies two conditions above
• Also, it suggests some applications using Bayes rule

5. Table of Contents
• Brief Introduction
• Dirichlet Process
• Applications

6. Dirichlet Process
• There are several ways to formulate Dirichlet Process
1. Dirichlet Distribution approach (with Kolmogorov
Consistency Theorem)
2. Gamma Process approach
3. Pólya (Blackwell-MacQueen) urn scheme (Chinese
Restraunt Process for special case)
4. Stick Breaking Construction (with de Finetti’s
Theorem)

7. Dirichlet Process
• There are several ways to formulate Dirichlet Process
1. Dirichlet Distribution approach (with Kolmogorov
Consistency Theorem)
2. Gamma Process approach
3. Pólya (Blackwell-MacQueen) urn scheme (Chinese
Restraunt Process for special case)
4. Stick Breaking Construction (with de Finetti’s
Theorem)

9. Kolmogorov Consistency Theorem
• Sum of two stochastic processes should consistently
construct another stochastic process
• Naively speaking, Kolmogorov Consistency Theorem
states that for any (mild) finite-dimensional distributions,
there exists a corresponding stochastic process

20. Dirichlet Process
• There are several ways to formulate Dirichlet Process
1. Dirichlet Distribution approach (with Kolmogorov
Consistency Theorem)
2. Gamma Process approach
3. Pólya (Blackwell-MacQueen) urn scheme (Chinese
Restraunt Process for special case)
4. Stick Breaking Construction (with de Finetti’s
Theorem)

25. Dirichlet Process
• There are several ways to formulate Dirichlet Process
1. Dirichlet Distribution approach (with Kolmogorov
Consistency Theorem)
2. Gamma Process approach
3. Pólya (Blackwell-MacQueen) urn scheme (Chinese
Restraunt Process for special case)
4. Stick Breaking Construction (with de Finetti’s
Theorem)

29. Dirichlet Process
• There are several ways to formulate Dirichlet Process
1. Dirichlet Distribution approach (with Kolmogorov
Consistency Theorem)
2. Gamma Process approach
3. Pólya (Blackwell-MacQueen) urn scheme (Chinese
Restraunt Process for special case)
4. Stick Breaking Construction (with de Finetti’s
Theorem)

32. Table of Contents
• Brief Introduction
• Dirichlet Process
• Applications

48. End of Document