automated learning with a probabilistic programming
play

Automated learning with a probabilistic programming language: Birch - PowerPoint PPT Presentation

Jan 08, 2024 •11 likes •1.43k views

Lawrence Murray Department of Information Technology, Uppsala University Outline Automated learning with a probabilistic programming language: Birch 1. The Birch probabilistic programming language. 2. The delayed sampling heuristic. 3. A

  1. Code Checkpoint observe y[t] Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  2. Code Checkpoint assume x[t] Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  3. Code Checkpoint observe y[t] Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  4. observe y[t] Checkpoint Code Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 2 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  5. observe y[t] Checkpoint Code Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 2 x[1] x[2] x[3] x[4] x[5] 2 y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  6. observe y[t] Checkpoint Code Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 3 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  7. Checkpoint assume x[t] Code Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 3 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  8. observe y[t] Checkpoint Code Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 3 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  9. observe y[t] Checkpoint Code Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 3 3 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  10. observe y[t] Checkpoint Code Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 3 3 x[1] x[2] x[3] x[4] x[5] 3 y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  11. observe y[t] Checkpoint Code Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 3 4 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  12. Checkpoint assume x[t] Code Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 3 4 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  13. observe y[t] Checkpoint Code Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 3 4 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  14. observe y[t] Checkpoint Code Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 3 4 4 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  15. observe y[t] Checkpoint Code Example #2 x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 3 4 4 x[1] x[2] x[3] x[4] x[5] 4 y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  16. observe y[t] Checkpoint Code Example #2: Kalman Filter x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 3 4 5 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  17. value x[1] Checkpoint Code Example #2: Kalman Filter x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 3 5 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  18. value x[1] Checkpoint Code Example #2: Kalman Filter x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 2 5 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  19. Code Checkpoint value x[1] Example #2: Kalman Filter x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 1 5 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  20. Code Checkpoint value x[1] Example #2: Kalman Filter x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); 5 x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  21. Code Checkpoint value x[1] Example #2: Kalman Filter x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  22. Checkpoint Code Example #2: Kalman Filter x[1] ~ Gaussian(0.0, 1.0); y[1] ~ Gaussian(x[1], 1.0); for (t in 2..T) { x[t] ~ Gaussian(a*x[t - 1], 1.0); y[t] ~ Gaussian(x[t], 1.0); } stdout.print(x[1]); x[1] x[2] x[3] x[4] x[5] y[1] y[2] y[3] y[4] y[5] Lawrence Murray 5 / 24

  23. Example #3 Lawrence Murray 6 / 24

  24. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  25. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  26. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  27. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  28. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  29. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  30. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  31. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  32. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  33. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  34. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  35. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  36. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  37. Example #3 1 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  38. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  39. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  40. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  41. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  42. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  43. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 1 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  44. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 1 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] 1 y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  45. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  46. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  47. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  48. Example #3 2 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  49. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  50. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  51. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  52. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  53. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  54. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 2 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  55. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 2 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] 2 y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  56. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 3 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  57. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 3 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  58. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 3 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  59. Example #3 3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 3 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  60. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 3 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  61. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 3 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  62. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 3 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  63. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 3 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

  64. Example #3 x_n[1] x_n[2] x_n[3] x_n[4] x_n[5] 1 2 3 x_l[1] x_l[2] x_l[3] x_l[4] x_l[5] y_l[1] y_l[2] y_l[3] y_l[4] y_l[5] y_n[1] y_n[2] y_n[3] y_n[4] y_n[5] Lawrence Murray 6 / 24

Recommend

CS 4110 Probabilistic Programming Probabilistic Programming It's not about writing software.

CS 4110 Probabilistic Programming Probabilistic Programming It's not about writing software.

CS 4110 Probabilistic Programming Probabilistic Programming It's not about writing software. Probabilistic Programming Probabilistic programming is a tool for statistical modeling. OR A probabilistic programming language is a plain old

688 views • 41 slides
Edward: Deep Probabilistic Programming Extended Seminar Systems and Machine Learning Steven

Edward: Deep Probabilistic Programming Extended Seminar Systems and Machine Learning Steven

Edward: Deep Probabilistic Programming Extended Seminar Systems and Machine Learning Steven Lang 13.02.2020 1 Outline Introduction Refresher on Probabilistic Modeling Deep Probabilistic Programming Compositional Representations in

912 views • 58 slides
Discrete Probabilistic Programming from First Principles Guy Van den Broeck The Fourth

Discrete Probabilistic Programming from First Principles Guy Van den Broeck The Fourth

Discrete Probabilistic Programming from First Principles Guy Van den Broeck The Fourth International Workshop on Declarative Learning Based Programming (DeLBP) Aug 11, 2019 What are probabilistic programs? What is the formal semantics? How

1.06k views • 63 slides
Reactive Probabilistic Programming Semantics with Mixed Nondeterministic/Probabilistic Automata

Reactive Probabilistic Programming Semantics with Mixed Nondeterministic/Probabilistic Automata

Reactive Probabilistic Programming Semantics with Mixed Nondeterministic/Probabilistic Automata Albert Benveniste Jean-Baptiste Raclet INRIA Rennes and IRIT Toulouse September 18, 2020 What is Probabilistic Programming? Bringing the

855 views • 59 slides
Democratizing Machine Learning and Artificial Intelligence: Probabilistic Programming with Scala

Democratizing Machine Learning and Artificial Intelligence: Probabilistic Programming with Scala

Democratizing Machine Learning and Artificial Intelligence: Probabilistic Programming with Scala Brian Ruttenberg, PhD Charles River Analytics bruttenberg@cra.com Goals of This Talk Introduce basic modeling concepts in Machine Learning and

862 views • 73 slides
Inference and Learning for Probabilistic Logic Programming Fabrizio Riguzzi Dipartimento di

Inference and Learning for Probabilistic Logic Programming Fabrizio Riguzzi Dipartimento di

Inference and Learning for Probabilistic Logic Programming Fabrizio Riguzzi Dipartimento di Matematica e Informatica Universit di Ferrara F. Riguzzi (DMI) Inference and learning for PLP 1 / 25 Outline Probabilistic Logic Languages 1

405 views • 25 slides
A Brief Introduction to Probabilistic and Quantum Programming Part II Ugo Dal Lago Universidade

A Brief Introduction to Probabilistic and Quantum Programming Part II Ugo Dal Lago Universidade

A Brief Introduction to Probabilistic and Quantum Programming Part II Ugo Dal Lago Universidade do Minho, Friday May 5, 2017 Section 1 Probabilistic Programming Languages Flipping a Coin Making any programming language probabilistic is

1.03k views • 92 slides
Probabilistic Programming for Bayesian Machine Learning Luke Ong University of Oxford

Probabilistic Programming for Bayesian Machine Learning Luke Ong University of Oxford

Probabilistic Programming for Bayesian Machine Learning Luke Ong University of Oxford 1 What is Machine Learning? Many related terms: neural networks, pattern recognition, data mining, data science, statistical modelling, AI, machine

990 views • 56 slides
Discrete Probabilistic Programming from First Principles Guy Van den Broeck The 6 th Workshop on

Discrete Probabilistic Programming from First Principles Guy Van den Broeck The 6 th Workshop on

Discrete Probabilistic Programming from First Principles Guy Van den Broeck The 6 th Workshop on Probabilistic Logic Programming (PLP) Sep 21, 2019 What are probabilistic programs? What is the formal semantics? How to do exact inference? What

830 views • 61 slides
Introduction to Probabilistic and Quantum Programming Part II Ugo Dal Lago BISS 2014, Bertinoro

Introduction to Probabilistic and Quantum Programming Part II Ugo Dal Lago BISS 2014, Bertinoro

Introduction to Probabilistic and Quantum Programming Part II Ugo Dal Lago BISS 2014, Bertinoro Section 1 Probabilistic Programming Languages Flipping a Coin Adding probabilistic behaviour to any programming language is relatively easy ,

459 views • 24 slides
Wolfe Practical Machine Learning Using Probabilistic Programming and Optimization Sameer Singh

Wolfe Practical Machine Learning Using Probabilistic Programming and Optimization Sameer Singh

Wolfe Practical Machine Learning Using Probabilistic Programming and Optimization Sameer Singh University of Washington Sebastian Riedel Tim Rocktschel Luke Hewitt University College London Large-scale NLP Information Extraction

331 views • 32 slides
Probabilistic Logic Programming and its Applications Luc De Raedt with many slides from Angelika

Probabilistic Logic Programming and its Applications Luc De Raedt with many slides from Angelika

Probabilistic Logic Programming and its Applications Luc De Raedt with many slides from Angelika Kimmig The Turing, London, September 11, 2017 1 A key question in AI: Dealing with uncertainty Reasoning with relational data ? Learning 2

2.19k views • 188 slides
Tutorial on Probabilistic Programming in Machine Learning Frank Wood Play Along 1. Download

Tutorial on Probabilistic Programming in Machine Learning Frank Wood Play Along 1. Download

Tutorial on Probabilistic Programming in Machine Learning Frank Wood Play Along 1. Download and install Leiningen http://leiningen.org/ 2. Fork and clone the Anglican Examples repository git@bitbucket.org:fwood/anglican-examples.git

806 views • 53 slides
Theres Something About Bayes Effective Probabilistic Programming for the Rest of Us James

Theres Something About Bayes Effective Probabilistic Programming for the Rest of Us James

Theres Something About Bayes Effective Probabilistic Programming for the Rest of Us James Bornholt Todd Mytkowicz Microsoft Research Kathryn S. McKinley Programs are doing (probabilistic) inference, even if they dont realise it.

548 views • 13 slides
Approximated Probabilistic Answer Set Programming Eduardo Menezes de Morais Department of

Approximated Probabilistic Answer Set Programming Eduardo Menezes de Morais Department of

Motivation ASP PSAT Probabilistic Answer Set Programming Resolution Methods Approximations Future work Approximated Probabilistic Answer Set Programming Eduardo Menezes de Morais Department of Computer Science Universidade de S ao

808 views • 48 slides
Automated Guidance of Post- Operative DBS Programming Webinar Will Begin at 12:00 PM EDT Outline

Automated Guidance of Post- Operative DBS Programming Webinar Will Begin at 12:00 PM EDT Outline

Automated Guidance of Post- Operative DBS Programming Webinar Will Begin at 12:00 PM EDT Outline Introduction to DBS Challenges and Technological Opportunities Kinesia ProView Automated, Technology-Assisted DBS Programming Deep

527 views • 30 slides
The Joy of Probabilistic Answer Set Programming Fabio G. Cozman Universidade de S ao Paulo

The Joy of Probabilistic Answer Set Programming Fabio G. Cozman Universidade de S ao Paulo

The Joy of Probabilistic Answer Set Programming Fabio G. Cozman Universidade de S ao Paulo Goal: to show that the credal semantics for Probabilistic Answer Set Programming (PASP) leads to a very useful modeling language. Answer set

375 views • 14 slides
MLE/MAP Matt Gormley Lecture 20 Oct 29, 2018 1 Q&A 9 PROBABILISTIC LEARNING 11

MLE/MAP Matt Gormley Lecture 20 Oct 29, 2018 1 Q&A 9 PROBABILISTIC LEARNING 11

10-601 Introduction to Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University MLE/MAP Matt Gormley Lecture 20 Oct 29, 2018 1 Q&A 9 PROBABILISTIC LEARNING 11 Probabilistic Learning

587 views • 44 slides
Probabilistic Graphical Models Probabilistic Graphical Models Parameter learning in Bayesian

Probabilistic Graphical Models Probabilistic Graphical Models Parameter learning in Bayesian

Probabilistic Graphical Models Probabilistic Graphical Models Parameter learning in Bayesian networks Siamak Ravanbakhsh Fall 2019 Learning objectives Learning objectives likelihood function and MLE role of the sufficient statistics MLE for

644 views • 60 slides
Probabilistic Programming Fun but Intricate Too! Joost-Pieter Katoen with Friedrich Gretz, Nils

Probabilistic Programming Fun but Intricate Too! Joost-Pieter Katoen with Friedrich Gretz, Nils

Probabilistic Programming is Fun, but Intricate Too Probabilistic Programming Fun but Intricate Too! Joost-Pieter Katoen with Friedrich Gretz, Nils Jansen, Benjamin Kaminski Christoph Matheja, Federico Olmedo and Annabelle McIver Mysore

775 views • 77 slides
Fine-Grained Semantics for Probabilistic Programs Benjamin Timon Martin Bichsel Gehr Vechev

Fine-Grained Semantics for Probabilistic Programs Benjamin Timon Martin Bichsel Gehr Vechev

Fine-Grained Semantics for Probabilistic Programs Benjamin Timon Martin Bichsel Gehr Vechev Probabilistic models Security Networking Machine Learning Robotics Randomized Algorithms Probabilistic programming Probability density

624 views • 36 slides
Programming Environments 30K (Eclipse: 118 MB - 4,000x as big) Integration - Foto von A Tool

Programming Environments 30K (Eclipse: 118 MB - 4,000x as big) Integration - Foto von A Tool

The Future of Programming Environments: Integration, Synergy, and Assistance Andreas Zeller, Saarland University Modern programming environments foster the integration of automated, Learning from Software extensible, and reusable tools. New

259 views • 21 slides
Probabilistic Graphical Models Probabilistic Graphical Models Structure learning in Bayesian

Probabilistic Graphical Models Probabilistic Graphical Models Structure learning in Bayesian

Probabilistic Graphical Models Probabilistic Graphical Models Structure learning in Bayesian networks Siamak Ravanbakhsh Fall 2019 Learning objectives Learning objectives why structure learning is hard? two approaches to structure learning

1.04k views • 74 slides
Machine Learning Probabilistic KNN. Mark Girolami girolami@dcs.gla.ac.uk Department of

Machine Learning Probabilistic KNN. Mark Girolami girolami@dcs.gla.ac.uk Department of

Machine Learning Probabilistic KNN. Mark Girolami girolami@dcs.gla.ac.uk Department of Computing Science University of Glasgow Probabilistic KNN June 21, 2007 p. 1/3 Probabilistic KNN KNN is a remarkably simple algorithm with proven

735 views • 35 slides

More recommend