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Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What do noisy datapoints tell us about the true signal? C. R. Hogg November 16, 2011 1/31

  1. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What is Bayesian Analysis? Essence of Bayes: • 2 questions for every guess (i.e. every θ ) 1. How likely does it make the actual data ? “LIKELIHOOD” 2. How plausible is it? “PRIOR” • Combine them to answer the main question : 1. What is your new probability, now that you’ve seen the data? “POSTERIOR” (Rev. Thomas Bayes, c. 1701 – 1761) 6/31

  2. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What is Bayesian Analysis? Essence of Bayes: • 2 questions for every guess (i.e. every θ ) 1. How likely does it make the actual data ? “LIKELIHOOD” 2. How plausible is it? “PRIOR” • Combine them to answer the main question : 1. What is your new probability, now that you’ve seen the data? “POSTERIOR” (Rev. Thomas Bayes, c. 1701 – 1761) 6/31

  3. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Likelihood of function p ( y | f ) • Example: artificial dataset 40 • Noise model: Poisson Intensity (number of counts) p ( y | f ) = f y e − f y ! 20 • Assume independent pixels 0 0.0 0.5 1.0 Scattering vector Q (1/A) 7/31

  4. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Likelihood of function p ( y | f ) • Example: artificial dataset 40 • Noise model: Poisson Intensity (number of counts) p ( y | f ) = f y e − f y ! 20 • Assume independent pixels 0 0.0 0.5 1.0 Scattering vector Q (1/A) 7/31

  5. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Likelihood of function p ( y | f ) • Example: artificial dataset 40 • Noise model: Poisson Intensity (number of counts) p ( y | f ) = f y e − f y ! 20 • Assume independent pixels 0 0.0 0.5 1.0 Scattering vector Q (1/A) 7/31

  6. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Likelihood of function p ( y | f ) • Example: artificial dataset 40 • Noise model: Poisson Intensity (number of counts) p ( y | f ) = f y e − f y ! 20 • Assume independent pixels 0 0.0 0.5 1.0 Scattering vector Q (1/A) 7/31

  7. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Likelihood of function p ( y | f ) • Example: artificial dataset 40 • Noise model: Poisson Intensity (number of counts) p ( y | f ) = f y e − f y ! 20 • Assume independent pixels • Problem: not plausible • (What makes a function 0 “plausible”?) 0.0 0.5 1.0 Scattering vector Q (1/A) 7/31

  8. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . “Plausibility” of function p ( f ) • Assume smooth and continuous • No functional form assumed 8/31

  9. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . “Plausibility” of function p ( f ) 40 Intensity (number of counts) • Assume smooth and continuous • No functional form 20 assumed 0 0.0 0.5 1.0 Scattering vector Q (1/A) 8/31

  10. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . “Plausibility” of function p ( f ) 40 Intensity (number of counts) • Assume smooth and continuous • No functional form 20 assumed 0 0.0 0.5 1.0 Scattering vector Q (1/A) 8/31

  11. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . “Plausibility” of function p ( f ) 40 Intensity (number of counts) • Assume smooth and continuous • No functional form 20 assumed 0 0.0 0.5 1.0 Scattering vector Q (1/A) 8/31

  12. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . “Plausibility” of function p ( f ) 40 Intensity (number of counts) • Assume smooth and continuous • No functional form 20 assumed 0 0.0 0.5 1.0 Scattering vector Q (1/A) 8/31

  13. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . “Plausibility” of function p ( f ) 40 Intensity (number of counts) • Assume smooth and continuous • No functional form 20 assumed 0 0.0 0.5 1.0 Scattering vector Q (1/A) 8/31

  14. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . “Plausibility” of function p ( f ) 40 Intensity (number of counts) • Assume smooth and continuous • No functional form 20 assumed 0 0.0 0.5 1.0 Scattering vector Q (1/A) 8/31

  15. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . “Plausibility” of function p ( f ) 40 Intensity (number of counts) • Assume smooth and continuous • No functional form 20 assumed • Naturally: unrelated to data 0 0.0 0.5 1.0 Scattering vector Q (1/A) 8/31

  16. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Posterior probability p ( f | y ) 40 • “Best of both worlds”: Intensity (number of counts) a Plausible curves, which b fit the data 20 0 0.0 0.5 1.0 Scattering vector Q (1/A) 9/31

  17. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Posterior probability p ( f | y ) 40 • “Best of both worlds”: Intensity (number of counts) a Plausible curves, which b fit the data 20 0 0.0 0.5 1.0 Scattering vector Q (1/A) 9/31

  18. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Posterior probability p ( f | y ) 40 • “Best of both worlds”: Intensity (number of counts) a Plausible curves, which b fit the data 20 0 0.0 0.5 1.0 Scattering vector Q (1/A) 9/31

  19. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Posterior probability p ( f | y ) 40 • “Best of both worlds”: Intensity (number of counts) a Plausible curves, which b fit the data 20 0 0.0 0.5 1.0 Scattering vector Q (1/A) 9/31

  20. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Posterior probability p ( f | y ) 40 • “Best of both worlds”: Intensity (number of counts) a Plausible curves, which b fit the data 20 0 0.0 0.5 1.0 Scattering vector Q (1/A) 9/31

  21. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Posterior probability p ( f | y ) 40 • “Best of both worlds”: Intensity (number of counts) a Plausible curves, which b fit the data 20 • To represent uncertainty: show many guesses • (Or, summarize them. . . ) 0 0.0 0.5 1.0 Scattering vector Q (1/A) 9/31

  22. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Posterior probability p ( f | y ) Quantitative uncertainty visuals Noisy Data 40 35 Intensity (no. of counts) 30 25 20 15 10 5 0 0 0.2 0.4 0.6 0.8 1 Q (1/A) 9/31

  23. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Posterior probability p ( f | y ) Quantitative uncertainty visuals Noisy Data 40 True Curve! 35 Intensity (no. of counts) 30 25 20 15 10 5 0 0 0.2 0.4 0.6 0.8 1 Q (1/A) 9/31

  24. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recap: Bayesian denoising Plausible curves 40 Intensity (number of counts) 20 0 0.0 0.5 1.0 Scattering vector Q (1/A) 10/31

  25. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recap: Bayesian denoising Curves which fit the data 40 Intensity (number of counts) 20 0 0.0 0.5 1.0 Scattering vector Q (1/A) 10/31

  26. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recap: Bayesian denoising Plausible curves which fit the data 40 Intensity (number of counts) 20 0 0.0 0.5 1.0 Scattering vector Q (1/A) 10/31

  27. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity • calculate probabilities for its values 11/31

  28. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , • calculate probabilities for its values • take “random draws” (roll the die, flip the coin. . . ) 11/31

  29. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , • calculate probabilities for its values • take “random draws” (roll the die, flip the coin. . . ) 11/31

  30. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , • calculate probabilities for its values • take “random draws” (roll the die, flip the coin. . . ) 11/31

  31. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , , • calculate probabilities for its values • take “random draws” (roll the die, flip the coin. . . ) 11/31

  32. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , , , • calculate probabilities for its values • take “random draws” (roll the die, flip the coin. . . ) 11/31

  33. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , , , • calculate probabilities for its values , • take “random draws” (roll the die, flip the coin. . . ) 11/31

  34. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , , , • calculate probabilities for its values , , • take “random draws” (roll the die, flip the coin. . . ) 11/31

  35. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , , , • calculate probabilities for its values , , , • take “random draws” (roll the die, flip the coin. . . ) 11/31

  36. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , , , • calculate probabilities for its values , , , , . . . • take “random draws” (roll the die, flip the coin. . . ) 11/31

  37. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , , , • calculate probabilities for its values , , , , . . . • take “random draws” (roll the die, flip the coin. . . ) • Random function F ( x ) ? F(x) 11/31 x

  38. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , , , • calculate probabilities for its values , , , , . . . • take “random draws” (roll the die, flip the coin. . . ) • Random function F ( x ) ? F(x) 11/31 x

  39. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , , , • calculate probabilities for its values , , , , . . . • take “random draws” (roll the die, flip the coin. . . ) • Random function F ( x ) ? F(x) 11/31 x

  40. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , , , • calculate probabilities for its values , , , , . . . • take “random draws” (roll the die, flip the coin. . . ) • Random function F ( x ) ? F(x) 11/31 x

  41. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , , , • calculate probabilities for its values , , , , . . . • take “random draws” (roll the die, flip the coin. . . ) • Random function F ( x ) ? F(x) 11/31 x

  42. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , , , • calculate probabilities for its values , , , , . . . • take “random draws” (roll the die, flip the coin. . . ) • Random function F ( x ) ? F(x) 11/31 x

  43. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , , , • calculate probabilities for its values , , , , . . . • take “random draws” (roll the die, flip the coin. . . ) • Random function F ( x ) ? F(x) 11/31 x

  44. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random variables; Random functions 1 1 1 1 1 p = 1 6 6 6 6 6 6 • Random variable F : an uncertain quantity F = , , , , , • calculate probabilities for its values , , , , . . . • take “random draws” (roll the die, flip the coin. . . ) • Random function F ( x ) ? F(x) 11/31 x

  45. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to think about “random functions”? • Function: a collection of individual values F(x) 12/31

  46. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to think about “random functions”? • Function: a collection of individual values F(x) 12/31

  47. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to think about “random functions”? • Function: a collection of individual values • Every value is a random variable, with. . . 1. variance 2. correlation F(x) 12/31

  48. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to think about “random functions”? • Function: a collection of individual values • Every value is a random variable, with. . . 1. variance 2. correlation F(x) 12/31

  49. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to think about “random functions”? • Function: a collection of individual values • Every value is a random variable, with. . . 1. variance 2. correlation F(x) 12/31

  50. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to think about “random functions”? • Function: a collection of individual values • Every value is a random variable, with. . . 1. variance 2. correlation F(x) 12/31

  51. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to think about “random functions”? • Function: a collection of individual values • Every value is a random variable, with. . . 1. variance 2. correlation F(x) 12/31

  52. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to think about “random functions”? • Function: a collection of individual values • Every value is a random variable, with. . . 1. variance 2. correlation F(x) 12/31

  53. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to think about “random functions”? • Function: a collection of individual values • Every value is a random variable, with. . . 1. variance 2. correlation F(x) 12/31

  54. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to think about “random functions”? • Function: a collection of individual values • Every value is a random variable, with. . . 1. variance 2. correlation F(x) 12/31

  55. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to think about “random functions”? • Function: a collection of individual values • Every value is a random variable, with. . . 1. variance 2. correlation • co rrelation × variance : covariance F(x) 12/31

  56. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to think about “random functions”? • Function: a collection of individual values • Every value is a random variable, with. . . 1. variance 2. correlation • co rrelation × variance : covariance • Gaussian Process: F(x) • Every point is a Random Variable • Any (finite) subset has Gaussian joint distribution 12/31

  57. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to read a Covariance Matrix • How to read the matrix? 0 Cov 2500 100 1 0 2 -100 0 3 0 1 2 3 0 1 2 3 X X 13/31

  58. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to read a Covariance Matrix • How to read the matrix? 1. By individual entries 0 Cov 2500 100 1 0 2 -100 0 3 0 1 2 3 0 1 2 3 X X 13/31

  59. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to read a Covariance Matrix • How to read the matrix? 1. By individual entries 0 Cov 2500 100 1 0 2 -100 0 3 0 1 2 3 0 1 2 3 X X 13/31

  60. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to read a Covariance Matrix • How to read the matrix? 1. By individual entries 0 Cov 2500 100 1 0 2 -100 0 3 0 1 2 3 0 1 2 3 X X 13/31

  61. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to read a Covariance Matrix • How to read the matrix? 1. By individual entries 0 Cov 2500 2. As a whole 100 1 (central stripe) 0 • Intensity: 2 height of features -100 0 3 • Width: 0 1 2 3 0 1 2 3 X X width of features 13/31

  62. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to read a Covariance Matrix 0 Stripe Intensity = Feature Height 100 1 0 2 • How to read the matrix? -100 3 0 1 2 3 0 1 2 3 X X 1. By individual entries 0 Cov 2500 2. As a whole 100 1 (central stripe) 0 • Intensity: 2 height of features -100 0 3 • Width: 0 1 2 3 0 1 2 3 X X width of features 13/31

  63. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to read a Covariance Matrix 0 Stripe Intensity = Feature Height 100 1 0 2 • How to read the matrix? -100 3 0 1 2 3 0 1 2 3 X X 1. By individual entries 0 Cov 2500 2. As a whole 100 1 (central stripe) 0 • Intensity: 2 height of features -100 0 3 • Width: 0 1 2 3 0 1 2 3 X X width of features 0 Stripe Width = Feature Width 100 1 0 2 -100 3 0 1 2 3 0 1 2 3 X X 13/31

  64. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Hydrocarbon combustion (Dave Sheen and Wing Tsang, NIST, Div. 632) Hydrocarbon burning simulations • Need (many!) reaction rate constants • Measured individually • Predictions are precise, quantitative 14/31

  65. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Hydrocarbon combustion (Dave Sheen and Wing Tsang, NIST, Div. 632) Hydrocarbon burning simulations • Need (many!) reaction rate constants • Measured individually • Predictions are precise, quantitative, wrong 14/31

  66. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrocarbon combustion: flame speed experiments • Datapoints (from several experiments) Flame speed 40 30 Speed (cm/s) 20 10 0 0.6 0.8 1.0 1.2 1.4 1.6 Fuel-to-oxygen ratio 15/31

  67. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrocarbon combustion: flame speed experiments • Datapoints (from several experiments) • Model : lengthscales Flame speed ℓ and σ f 40 30 Speed (cm/s) 20 10 0 0.6 0.8 1.0 1.2 1.4 1.6 Fuel-to-oxygen ratio 15/31

  68. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrocarbon combustion: flame speed experiments • Datapoints (from several experiments) • Model : lengthscales Flame speed ℓ and σ f 40 • ± 1 σ range 30 Speed (cm/s) 20 10 0 0.6 0.8 1.0 1.2 1.4 1.6 Fuel-to-oxygen ratio 15/31

  69. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrocarbon combustion: flame speed experiments • Datapoints (from several experiments) • Model : lengthscales Flame speed ℓ and σ f 40 • ± 1 σ range 30 Speed (cm/s) • See also: individual curves 20 10 0 0.6 0.8 1.0 1.2 1.4 1.6 Fuel-to-oxygen ratio 15/31

  70. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrocarbon combustion: flame speed experiments • Datapoints (from several experiments) • Model : lengthscales Flame speed ℓ and σ f 40 • ± 1 σ range 30 Speed (cm/s) • See also: individual curves 20 • But where did this model come from. . . ? 10 0 0.6 0.8 1.0 1.2 1.4 1.6 Fuel-to-oxygen ratio 15/31

  71. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Occam’s razor, intro • William of Occam c. 1288 - c. 1348 • Gave us Occam’s Razor 16/31

  72. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Occam’s razor, intro • William of Occam c. 1288 - c. 1348 • Gave us Occam’s Razor • (slightly paraphrased in the name of science) 16/31

  73. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Occam’s razor, intro • William of Occam c. 1288 - c. 1348 • Gave us Occam’s Razor • (slightly paraphrased in the name of science) • Claim: use probability , get this automatically • (And, quantitative , too!) 16/31

  74. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Occam’s razor, intro • William of Occam c. 1288 - c. 1348 • Gave us Occam’s Razor MODEL 1: • (slightly paraphrased in flat the name of science) • Claim: use probability , get this automatically • (And, quantitative , too!) • Example: 3 models. . . 16/31

  75. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Occam’s razor, intro • William of Occam c. 1288 - c. 1348 • Gave us Occam’s Razor MODEL 2: • (slightly paraphrased in flat + line the name of science) • Claim: use probability , get this automatically • (And, quantitative , too!) • Example: 3 models. . . 16/31

  76. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Occam’s razor, intro • William of Occam c. 1288 - c. 1348 • Gave us Occam’s Razor MODEL 3: • (slightly paraphrased in flat + line + wiggle the name of science) • Claim: use probability , get this automatically • (And, quantitative , too!) • Example: 3 models. . . 16/31

  77. Overview Bayesian Analysis Gaussian Processes Scattering Curves Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Occam’s razor in action • Some models can explain more datasets 17/31

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