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## Stick-Breaking Constructions

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**Stick-Breaking Constructions**Patrick Dallaire June 10th, 2011**Outline**• Introduction of the Stick-Breaking process**Outline**• Introduction of the Stick-Breaking process • Presentation of fundamental representation**Outline**• Introduction of the Stick-Breaking process • Presentation of fundamental representation • The Dirichlet process • The Pitman-Yor process • The Indian buffet process**Outline**• Introduction of the Stick-Breaking process • Presentation of fundamental representation • The Dirichlet process • The Pitman-Yor process • The Indian buffet process • Definition of the Beta process**Outline**• Introduction of the Stick-Breaking process • Presentation of fundamental representation • The Dirichlet process • The Pitman-Yor process • The Indian buffet process • Definition of the Beta process • A Stick-Breaking construction of Beta process**Outline**• Introduction of the Stick-Breaking process • Presentation of fundamental representation • The Dirichlet process • The Pitman-Yor process • The Indian buffet process • Definition of the Beta process • A Stick-Breaking construction of Beta process • Conclusion and current work**The Stick-Breaking process**• Assume a stick of unit length**The Stick-Breaking process**• Assume a stick of unit length**The Stick-Breaking process**• Assume a stick of unit length • At each iteration, a part of the remaining stick is broken by sampling the proportion to cut**The Stick-Breaking process**• Assume a stick of unit length • At each iteration, a part of the remaining stick is broken by sampling the proportion to cut**The Stick-Breaking process**• Assume a stick of unit length • At each iteration, a part of the remaining stick is broken by sampling the proportion to cut**The Stick-Breaking process**• Assume a stick of unit length • At each iteration, a part of the remaining stick is broken by sampling the proportion to cut**The Stick-Breaking process**• Assume a stick of unit length • At each iteration, a part of the remaining stick is broken by sampling the proportion to cut**The Stick-Breaking process**• Assume a stick of unit length • At each iteration, a part of the remaining stick is broken by sampling the proportion to cut**The Stick-Breaking process**• Assume a stick of unit length • At each iteration, a part of the remaining stick is broken by sampling the proportion to cut**The Stick-Breaking process**• Assume a stick of unit length • At each iteration, a part of the remaining stick is broken by sampling the proportion to cut**The Stick-Breaking process**• Assume a stick of unit length • At each iteration, a part of the remaining stick is broken by sampling the proportion to cut**The Stick-Breaking process**• Assume a stick of unit length • At each iteration, a part of the remaining stick is broken by sampling the proportion to cut**The Stick-Breaking process**• Assume a stick of unit length • At each iteration, a part of the remaining stick is broken by sampling the proportion to cut**The Stick-Breaking process**• Assume a stick of unit length • At each iteration, a part of the remaining stick is broken by sampling the proportion to cut • How should we sample these proportions?**Beta random proportions**• Let be the proportion to cut at iteration**Beta random proportions**• Let be the proportion to cut at iteration • The remaining length can be expressed as**Beta random proportions**• Let be the proportion to cut at iteration • The remaining length can be expressed as • Thus, the broken part is defined by**Beta random proportions**• Let be the proportion to cut at iteration • The remaining length can be expressed as • Thus, the broken part is defined by • We first consider the case where**Beta distribution**• The Beta distribution is a density function on • Parameters and control its shape**The Dirichlet process**• Dirichlet processes are often used to produce infinite mixture models**The Dirichlet process**• Dirichlet processes are often used to produce infinite mixture models • Each observation belongs to one of the infinitely many components**The Dirichlet process**• Dirichlet processes are often used to produce infinite mixture models • Each observation belongs to one of the infinitely many components • The model ensures that only a finite number of components have appreciable weight**The Dirichlet process**• A Dirichlet process, , can be constructed according to a Stick-Breaking process • Where is the base distribution and is a unit mass at**The Pitman-Yor process**• A Pitman-Yor process, , can be constructed according to a Stick-Breaking process • Where and