An introduction to bayesian statistics

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An introduction to bayesian statistics

  1. 1. AnIntroductionto BayesianStatisticsPaul HerendeenApril 2013
  2. 2. 020004000600080001960 1970 1980 1990 2000 2010WOS "Bayesian"Citations by YearThe rise of Bayesianstatistics
  3. 3. So…What are BayesianStatistics?
  4. 4. So…What are BayesianStatistics?1) A fundamentally different approach toprobability
  5. 5. So…What are BayesianStatistics?1) A fundamentally different approach toprobability2) An associated set of mathematical tools
  6. 6. Frequentists vs. BayesianRound 1ParametersfixedDatavariesDatafixedParametersVary
  7. 7. Frequentists vs. BayesianRound 1
  8. 8. Frequentists vs. BayesianRound 1ConfidenceIntervalCredibleInterval
  9. 9. Conditional Probabilityin 2 minutes
  10. 10. Conditional Probabilityin 2 minutesAll possible outcomes
  11. 11. Conditional Probabilityin 2 minutes
  12. 12. Conditional Probabilityin 2 minutes
  13. 13. Conditional Probabilityin 2 minutes
  14. 14. Conditional Probabilityin 2 minutes
  15. 15. Conditional Probabilityin 2 minutes
  16. 16. Bayes’ Theorem
  17. 17. Bayes’ TheoremPrior
  18. 18. Bayes’ TheoremLikelihoodPrior
  19. 19. Bayes’ TheoremLikelihoodPriorEvidence
  20. 20. Bayes’ TheoremLikelihoodPriorPosteriorEvidence
  21. 21. Frequentist vs. BayesianRound 2“The Strength of the Prior”
  22. 22. SparseData
  23. 23. AbundantData
  24. 24. UniformPrior
  25. 25. Where do Priors ComeFrom?
  26. 26. So…What are BayesianStatistics?1) A fundamentally different approach toprobability2) An associated set of mathematical tools
  27. 27. How do you actuallydo this?
  28. 28. So how do you actuallydo this?1. Analytical methods
  29. 29. So how do you actuallydo this?1. Analytical methods2. Grid approximation
  30. 30. So how do you actuallydo this?1. Analytical methods2. Grid approximation3. Markov Chain Monte Carlo
  31. 31. MCMC• Algorithm for exploring parameter space
  32. 32. MCMC• Algorithm for exploring parameter space1.Pick a starting point
  33. 33. MCMC• Algorithm for exploring parameter space1.Pick a starting point2.Propose a move
  34. 34. MCMC• Algorithm for exploring parameter space1.Pick a starting point2.Propose a move3.Accept or decline move based onprobability
  35. 35. MCMC• Algorithm for exploring parameter space1.Pick a starting point2.Propose a move3.Accept or decline move based onprobability• Time spent at each point approximatesparameter distribution
  36. 36. MCMC• Algorithm for exploring parameter space1.Pick a starting point2.Propose a move3.Accept or decline move based onprobability• Time spent at each point approximatesparameter distribution• E.g. Metropolis-Hastings, Gibbs sampling
  37. 37. MCMC2D example
  38. 38. MCMC2D example
  39. 39. So what does all thisget us?
  40. 40. Bayesian methodsreally shine in complex(hierarchical) models…
  41. 41. For example,IndividualFecundityGroupEffectPopulationEffectForagingsuccessEnvironment
  42. 42. or…IndividualFecundityGroupEffectPopulationEffectEnvironment
  43. 43. Many benefits to thisapproach• Simultaneously estimate parameters• …as well as parameter relationships• “Borrow” strength across studies• Model comparison
  44. 44. So, is it a BayesianRevolution?
  45. 45. Bayesian stats can domost thingsfrequentist,
  46. 46. Bayesian stats can domost thingsfrequentist, but…• Many simple models don‟t gain much• Better do something „boring‟ well thansomething exciting poorly
  47. 47. Bayesian stats can domost thingsfrequentist, but…• Many simple models don‟t gain much• Better do something „boring‟ well thansomething exciting poorly• Don‟t be this guy
  48. 48. DO use Bayesianmethods if• You have a complex model with manyinteracting parameters• You have „messy‟ data• You don‟t want to make assumptionsabout distributions
  49. 49. In Conclusion• Bayesian methods are powerful toolsfor ecological research• Like most things statistical, they areno substitute for thinking• They are here to stay, and you shouldat least be familiar with them
  50. 50. Great, I want to learnmore!JAGS(Just Another Gibbs Sampler)

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