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A	Bayesian	Analysis	of	Some	
Nonparametric	Problems
(Ferguson,	1973)
2016.07.04
Sangwoo	Mo
Table	of	Contents
• Brief	Introduction
• Dirichlet	Process
• Applications
Table	of	Contents
• Brief	Introduction
• Dirichlet	Process
• Applications
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
Table	of	Contents
• Brief	Introduction
• Dirichlet	Process
• Applications
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)
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)
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
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)
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)
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)
Table	of	Contents
• Brief	Introduction
• Dirichlet	Process
• Applications
End	of	Document

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Dirichlet Process