stick breaking constructions n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Stick-Breaking Constructions PowerPoint Presentation
Download Presentation
Stick-Breaking Constructions

play fullscreen
1 / 94

Stick-Breaking Constructions

106 Views Download Presentation
Download Presentation

Stick-Breaking Constructions

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

  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

  41. Construction demo

  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