1. Likelihood and Bayes Brian O’Meara http://www.brianomeara.info http://xkcd.com/1132/

2. Predict the number of heads in the next 18 flips (write prediction and your name on sticky note). Guess right and get a candy [regardless of how many others get it right]

3. Probability of getting a single heads given p = p

4. Probability of getting two heads given p = p 2

5. Probability(data given p) = p 2 Probability(data | p) = p 2 Likelihood(p | data) = Probability(data | p) = p 2

6. Probability of getting two heads given p = p 2 Likelihood Parameter p

7. Likelihood

8. Parameter p Likelihood

9. Abzhanov et al. 2006

10. Evolutionary rate Likelihood

11. Properties of likelihood as a method Consistent (given correct model) Efficient (no other estimator has a lower mean squared error as dataset size approaches infinity) Often biased, bias decreases with sample size

12. Likelihoods can get really small, even with simple fair coin Number of throws Number of heads LikelihoodLog Likelihood 110.5-0.6931472 520.3125-1.163151 100250.00000019314-15.46935 500150 0.00000000000 0000000005279 -44.38798 Limits of your machine's precision in R: noquote(unlist(format(.Machine)))

13. Likelihood ratio test Models must be nested (one must be a restriction of the other)

14. Huelsenbeck& Crandall 1997

15. AIC ∆AIC Level of empirical support for model 0 – 2Substantial 4 – 7Considerably less 10+Essentially none Burnham & Anderson 2004

16. Butler & King 2004 ∆AICc7.0311.039.484.470.00 relative likelihood 0.0300.0040.0000.1071 weight 0.030.000.080.090.87 Model averaged sigma: 0.21 x 0.03 + 0.21 x 0 + 0.20 x 0.08 + 0.47 x 0.09 + 0.22 x 0.87 = 0.26

17. Parameter p Likelihood How many of you guessed that 18 x 2/3 = 12 would be heads when candy was on the line?

18. http://xkcd.com/1236/ Conditional probability

19. http://xkcd.com/1236/ Bayesian statistics Hyp.Data Hyp.

20. Bayesian statistics P(H | D) = P(D | H) x P(H) P(D) P(H | D) = Posterior P(D | H) = Likelihood P(H) = Prior P(D) = Prob of the data, over any hypothesis

21. Bayesian statistics P(H | D) = P(D | H) x P(H) P(D) P(H | D) = Posterior P(D | H) = Likelihood P(H) = Prior P(D) = Prob of the data, over any hypothesis

22. LikelihoodPriorsPosteriors

23. Bayesian statistics P(H | D) = P(D | H) x P(H) P(D) P(H | D) = Posterior P(D | H) = Likelihood P(H) = Prior P(D) = Prob of the data, over any hypothesis

24. Markov Chain Monte Carlo Markov Chain: series of steps, each step ONLY depends on current state, not states further in the past Monte Carlo: repeated sampling from distribution. Think Las Vegas

25. http://salmagundiboston.blogspot.com/

26. Bayes Factors

27. Reversible jump MCMC Model 1 Model 2 Model 1

28. Approximate Bayesian Computation (can do with likelihood, too)

29. Multivariate normal

32. What is the probability of heads for our coin?

36. Let p = 0.2 Sim 1: HTTTTHTT Sim 2: TTHHTTTT Sim 3: TTTHHTHH Sim 4: TTTTTTTT Sim 5: TTTTTTTT Sim 6: THHTTTTH Sim 7: TTHHTTHT Sim 8: TTTTHHHT Sim 9: HTTHTTTT Sim 10: THHHTHTT

37. What is the probability of heads for our coin? Let p = 0.2 Sim 1: HTTTTHTT Sim 2: TTHHTTTT Sim 3: TTTHHTHH Sim 4: TTTTTTTT Sim 5: TTTTTTTT Sim 6: THHTTTTH Sim 7: TTHHTTHT Sim 8: TTTTHHHT Sim 9: HTTHTTTT Sim 10: THHHTHTT

38. What is the probability of heads for our coin? Let p = 0.2 Sim 1: HTTTTHTT Sim 2: TTHHTTTT Sim 3: TTTHHTHH Sim 4: TTTTTTTT Sim 5: TTTTTTTT Sim 6: THHTTTTH Sim 7: TTHHTTHT Sim 8: TTTTHHHT Sim 9: HTTHTTTT Sim 10: THHHTHTT P(data) ≈ 1/10

39. What is the probability of heads for our coin? 200 simulations per p True Approximation

40. What is the probability of heads for our coin? 2,000 simulations per p True Approximation

41. What is the probability of heads for our coin? 20,000 simulations per p True Approximation

42. What is the probability of heads for our coin? 200,000 simulations per p True Approximation