All About that Bayes: Probability, Statistics, and the Quest to Quantify Uncertainty
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The document discusses the use of probability to quantify uncertainty in statistics, emphasizing its historical development and application. It highlights notable figures such as Thomas Bayes and Pierre-Simon Laplace, while contrasting Bayesian and frequentist approaches to statistical inference. The presentation concludes with a summary of learned concepts and further reading recommendations.
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LLNL-PRES-697098
The
Sun
Will
Come
Out
Tomorrow?
What is the probability that the sun will rise
tomorrow?
E θ | X = x[ ]=
# of times sun has come up + 1
# of times sun has come up + 2](https://image.slidesharecdn.com/allaboutthatbayes072816outside-160920175359/85/All-About-that-Bayes-Probability-Statistics-and-the-Quest-to-Quantify-Uncertainty-39-320.jpg)
All About that Bayes: Probability, Statistics, and the Quest to Quantify Uncertainty
- 1. LLNL-PRES-697098 This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC All About That Bayes Probability, Sta4s4cs, and the Quest to Quan4fy Uncertainty Kris%n P. Lennox Director of Sta%s%cal Consul%ng July 28, 2016
- 2. 2 LLNL-PRES-697098 Man of the (Literal) Hour Probably not Thomas Bayes, but often mistaken for him Source: Wikipedia
- 3. 3 LLNL-PRES-697098 Central Dogma of Inferen4al Sta4s4cs Statisticians use probability to describe uncertainty.
- 4. 4 LLNL-PRES-697098 You Are Here Why This Matters What is Probability? What is Uncertainty? An Incomplete History of Uncertainty Quantification The BIG Reveal
- 5. 5 LLNL-PRES-697098 You Are Here Why This Matters What is Probability? What is Uncertainty? An Incomplete History of Uncertainty Quantification The BIG Reveal
- 6. 6 LLNL-PRES-697098 § Probability § Distribu%on § Parameter § Likelihood What is Probability? 1933 A. N. Kolmogorov Copyright MFO, Creative Commons License
- 7. 7 LLNL-PRES-697098 A. N. Kolmogorov Copyright MFO, Creative Commons License § Probability is a measure. § Distribu%on § Parameter § Likelihood What is Probability? 1933
- 8. 8 LLNL-PRES-697098 § Probability is a measure. § Distribu%ons define measure of events. § Parameter § Likelihood Exponential Normal/Gaussian What is Probability?
- 9. 9 LLNL-PRES-697098 § Probability is a measure. § Distribu%ons define measure of events. § Parameters define distribu%ons. § Likelihood Exponential Normal/Gaussian What is Probability?
- 10. 10 LLNL-PRES-697098 f (x) = Pr(X = x |Θ =θ) § Probability is a measure. § Distribu%ons define measure of events. § Parameters define distribu%ons. § Likelihood fixes data and varies parameters. What is Probability?
- 11. 11 LLNL-PRES-697098 § Probability is a measure. § Distribu%ons define measure of events. § Parameters define distribu%ons. § Likelihood fixes data and varies parameters. l(θ) = Pr(X = x |Θ =θ) What is Probability?
- 12. 12 LLNL-PRES-697098 You Are Here Why This Matters What is Probability? What is Uncertainty? An Incomplete History of Uncertainty Quantification The BIG Reveal
- 13. 13 LLNL-PRES-697098 A Fable The Sta4s4cal Lunch Bunch and the Summer Student Revolt of ‘15 1
- 14. 14 LLNL-PRES-697098 A Fable The Sta4s4cal Lunch Bunch and the Summer Student Revolt of ‘15 2
- 15. 15 LLNL-PRES-697098 A Fable The Sta4s4cal Lunch Bunch and the Summer Student Revolt of ‘15
- 16. 16 LLNL-PRES-697098 A Fable The Sta4s4cal Lunch Bunch and the Summer Student Revolt of ‘15
- 17. 17 LLNL-PRES-697098 A Fable The Sta4s4cal Lunch Bunch and the Summer Student Revolt of ‘15
- 18. 18 LLNL-PRES-697098 A Fable The Sta4s4cal Lunch Bunch and the Summer Student Revolt of ‘15 3
- 19. 19 LLNL-PRES-697098 A Fable The Sta4s4cal Lunch Bunch and the Summer Student Revolt of ‘15
- 20. 20 LLNL-PRES-697098 A Fable The Sta4s4cal Lunch Bunch and the Summer Student Revolt of ‘15
- 21. 21 LLNL-PRES-697098 A Fable The Sta4s4cal Lunch Bunch and the Summer Student Revolt of ‘15
- 22. 22 LLNL-PRES-697098 A Fable The Sta4s4cal Lunch Bunch and the Summer Student Revolt of ‘15 4
- 23. 23 LLNL-PRES-697098 A Fable The Sta4s4cal Lunch Bunch and the Summer Student Revolt of ‘15
- 24. 24 LLNL-PRES-697098 You Are Here Why This Matters What is Probability? What is Uncertainty? An Incomplete History of Uncertainty Quantification The BIG Reveal
- 25. 25 LLNL-PRES-697098 In the Beginning… 1654 P. Fermat Source: Wikipedia, Creative Commons License B. Pascal Source: Wikipedia, Creative Commons License
- 26. 26 LLNL-PRES-697098 Thomas Bayes and the Doctrine of Chances 1763 Still not Thomas Bayes Source: Wikipedia
- 27. 27 LLNL-PRES-697098 Blindfolded 1-‐Dimensional Table Bocce
- 28. 28 LLNL-PRES-697098 Blindfolded 1-‐Dimensional Table Bocce PriorDensity n=0
- 29. 29 LLNL-PRES-697098 Blindfolded 1-‐Dimensional Table Bocce PosteriorDensity n=1
- 30. 30 LLNL-PRES-697098 PosteriorDensity Blindfolded 1-‐Dimensional Table Bocce n=2
- 31. 31 LLNL-PRES-697098 PosteriorDensity Blindfolded 1-‐Dimensional Table Bocce n=10
- 32. 32 LLNL-PRES-697098 PosteriorDensity Blindfolded 1-‐Dimensional Table Bocce n=25
- 33. 33 LLNL-PRES-697098 § What is the probability that event x occurs given that event y occurs? Bayes Theorem Pr(X = x |Y = y) = Pr(Y = y | X = x)Pr(X = x) Pr(Y = y)
- 34. 34 LLNL-PRES-697098 What is the probability that our distribu,on parameter is θ given that we have observed data x? Bayes Theorem – Bayesian Version Pr(Θ =θ | X = x) = Pr(X = x |Θ =θ)Pr(Θ =θ) Pr(X = x) Prior distribution of θ Posterior distribution of θ given x A constant of integration that most people don’t talk about Likelihood
- 35. 35 LLNL-PRES-697098 The Man Who Invented Sta4s4cs 18th Century P. S. Laplace Source: Wikipedia, Creative Commons License
- 36. 36 LLNL-PRES-697098 The Prior and Its Enemies Pr(Θ =θ | X = x)∝ Pr(X = x |Θ =θ)Pr(Θ =θ) Posterior distribution of θ given x Likelihood Where does the prior come from? Prior distribution of θ
- 37. 37 LLNL-PRES-697098 Pr(Θ =θ | X = x)∝ Pr(X = x |Θ =θ)Pr(Θ =θ) Prior distribution of θPosterior distribution of θ given x Likelihood What is the probability that the sun will rise tomorrow? The Sun Will Come Out Tomorrow?
- 38. 38 LLNL-PRES-697098 Pr(Θ =θ | X = x)∝ Pr(X = x |Θ =θ) Posterior distribution of θ given x Likelihood What is the probability that the sun will rise tomorrow? The Sun Will Come Out Tomorrow? in (0,1)
- 39. 39 LLNL-PRES-697098 The Sun Will Come Out Tomorrow? What is the probability that the sun will rise tomorrow? E θ | X = x[ ]= # of times sun has come up + 1 # of times sun has come up + 2
- 40. 40 LLNL-PRES-697098 The Sun Will Come Out Tomorrow?* What is the probability that the sun will rise tomorrow? = 0.9999995= 5000×365.25 + 1 5000×365.25 + 2 *As calculated by Laplace
- 41. 41 LLNL-PRES-697098 The Frequen4sts Early 20th Century J. Neyman Source: Wikipedia, Creative Commons License R. A. Fisher Source: Wikipedia, Creative Commons License
- 42. 42 LLNL-PRES-697098 Case Study: Interval Es4ma4on What is lowest reasonable Pr(X=1)?
- 43. 43 LLNL-PRES-697098 Case Study: Interval Es4ma4on Bayesian Solu%on (Credible Interval) 1. Pick a prior.
- 44. 44 LLNL-PRES-697098 Case Study: Interval Es4ma4on Bayesian Solu%on (Credible Interval) 1. Pick a prior. 2. Calculate posterior.
- 45. 45 LLNL-PRES-697098 Bayesian Solu%on (Credible Interval) 1. Pick a prior. 2. Calculate posterior. 3. Find 5th percen%le. Case Study: Interval Es4ma4on 95% (subjec%ve) probability that Pr(X=1) is at least 15.3%.
- 46. 46 LLNL-PRES-697098 Case Study: Interval Es4ma4on Frequen%st Solu%on (Confidence Interval) 1. Determine all results that are as or more consistent with outcome of interest. Care about 12 or more 1s in 50 rolls.
- 47. 47 LLNL-PRES-697098 Frequen%st Solu%on (Confidence Interval) 1. Determine all results that are as or more consistent with outcome of interest. 2. Iden%fy all Pr(X=1) that have >5% chance of producing 12 or more 1s. With 95% confidence, Pr(X=1) is at least 14.5%. Case Study: Interval Es4ma4on Pr(12ormore1sin50rolls) Pr(X=1)
- 48. 48 LLNL-PRES-697098 Case Study: Interval Es4ma4on Both give Pr(X=1) >15%* with “uncertainty” of 5%. *15.3% (Bayesian with Jeffreys prior) vs. 14.5% (frequen%st) … but confidence interval takes a lot longer to explain.
- 49. 49 LLNL-PRES-697098 Why Did This Catch On? Objec%ve probability is restric%ve, but results mean the same thing to everyone. (Even if you don’t know what they mean.)
- 50. 50 LLNL-PRES-697098 BaXle of the Bayesians 20th Century -‐ ??? vs Representative likeness of a subjective BayesianRepresentative likeness of an objective Bayesian
- 51. 51 LLNL-PRES-697098 The Search For Scorpion 1968
- 52. 52 LLNL-PRES-697098 The Search For Scorpion 1968 Contact M8/3
- 53. 53 LLNL-PRES-697098 The Search For Scorpion 1968 Scorpion location
- 54. 54 LLNL-PRES-697098 Computa4on Late 20th Century
- 55. 55 LLNL-PRES-697098 You Are Here Why This Matters What is Probability? What is Uncertainty? An Incomplete History of Uncertainty Quantification The BIG Reveal
- 56. 56 LLNL-PRES-697098 My Uncertainty Quan4fica4on Toolbox
- 57. 57 LLNL-PRES-697098 What Have We Learned Today? Statisticians use probability to describe uncertainty. We do not always agree about how this should be done.
- 58. 58 LLNL-PRES-697098 Ques%ons? lennox@llnl.gov Thank you for coming! Thomas Bayes would never have worn these glasses.
- 59. 59 LLNL-PRES-697098 § The Theory That Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy. McGrayne, S. B. Yale University Press. (2011) § Bruno de FineJ: Radical Probabilist. Ed. Galavoj, M. C. Texts in Philosophy 8. College Publica%ons. (2009) § Fisher, Neyman, and the Crea,on of Classical Sta,s,cs. Lehmann, E. L. Springer. (2011) § The History of Probability and Sta,s,cs and Their Applica,ons Before 1750. Hald, A. Wiley-‐Interscience. (2003) § Founda,ons of the Theory of Probability. Kolmogorov, A. N. 2nd English Edi%on. Chelsea Publishing Company. (1950) § “Opera%ons Analysis During the Underwater Search for Scorpion.” Richardson, H. R. and Stone, L. D. Naval Research Quarterly. 18, pp. 141-‐157 (1971) § “An Essay Towards Solving a Problem in the Doctrine of Chances.” Bayes, T. and Price, R. Philosophical Transac,ons. 53, pp. 370-‐418 (1763) § “Sta%s%cal Analysis and the Illusion of Objec%vity.” Berger, J. and Berry, D. American Scien,st. 76, pp. 159-‐165 (1988) § “You May Believe You are a Bayesian but You are Probably Wrong.” Senn, S. Ra,onality, Markets and Morals. 2, pp. 48-‐66 (2011) § “The Case for Objec%ve Bayes.” Berger, J. Bayesian Analysis. 1, pp. 1-‐17 (2004) § “When Genius Errs: R.A. Fisher and the Lung Cancer Controversy.” Stoley, P. D. American Journal of Epidemiology. 133, pp. 416-‐425 (1991) § “The Evolu%on of Markov Chain Monte Carlo Methods.” Richey, M. The American Mathema,cal Monthly. 117, 383-‐413 (May 2010) § and of course Wikipedia.org Further Reading