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Stick-Breaking Constructions

Stick-Breaking Constructions. Patrick Dallaire June 10 th , 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

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Stick-Breaking Constructions

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  1. Stick-Breaking Constructions Patrick Dallaire June 10th, 2011

  2. Outline • Introduction of the Stick-Breaking process

  3. Outline • Introduction of the Stick-Breaking process • Presentation of fundamental representation

  4. Outline • Introduction of the Stick-Breaking process • Presentation of fundamental representation • The Dirichlet process • The Pitman-Yor process • The Indian buffet process

  5. 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

  6. 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

  7. 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

  8. The Stick-Breaking process

  9. The Stick-Breaking process • Assume a stick of unit length

  10. The Stick-Breaking process • Assume a stick of unit length

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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?

  23. Beta random proportions • Let be the proportion to cut at iteration

  24. Beta random proportions • Let be the proportion to cut at iteration • The remaining length can be expressed as

  25. 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

  26. 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

  27. Beta distribution • The Beta distribution is a density function on • Parameters and control its shape

  28. The Dirichlet process

  29. The Dirichlet process • Dirichlet processes are often used to produce infinite mixture models

  30. The Dirichlet process • Dirichlet processes are often used to produce infinite mixture models • Each observation belongs to one of the infinitely many components

  31. 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

  32. 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

  33. Construction demo

  34. Construction demo

  35. Construction demo

  36. Construction demo

  37. Construction demo

  38. Construction demo

  39. Construction demo

  40. Construction demo

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  42. Construction demo

  43. Construction demo

  44. Construction demo

  45. Construction demo

  46. Construction demo

  47. Construction demo

  48. Construction demo

  49. The Pitman-Yor process

  50. The Pitman-Yor process • A Pitman-Yor process, , can be constructed according to a Stick-Breaking process • Where and

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