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Probability distributions and likelihood

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Probability distributions and likelihood

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    1. Likelihood 1 Probability distributions and likelihood

    2. Likelihood 2

    3. Likelihood 3 Overview Probability distributions - binomial, poisson, normal, lognormal, negative binomial, beta Likelihood Likelihood profile The concept of support Model Selection Likelihood Ratio, AIC Robustness - contradictory data

    4. Likelihood 4 The binomial distribution discrete outcomes discrete trials Consider a discrete outcome - a coin is heads or tails, an animal (or plant) lives or dies We examine a fixed number of such events - a number of flips of the coin, a certain number of animals that may or may not survive

    5. Likelihood 5 The binomial formula

    6. Likelihood 6 Factorial term

    7. Likelihood 7 The Poisson outcomes discrete, continuous number of observations

    8. Likelihood 8 Limitations of Poisson Has only one parameter, which is both the mean and the variance We often have discrete count data, but want the variance to be estimable or at least larger than Poisson

    9. Likelihood 9 Thus we often use the negative binomial Also discrete outcomes with continuous observations Is derived from the Poisson where the rate parameter is a random variable

    10. Likelihood 10 The negative binomial outcomes discrete, continuous observations

    11. Likelihood 11 The normal distribution continuous distribution

    12. Likelihood 12 This is the familiar bell shaped curve

    13. Likelihood 13 Quiz: But what is the Y axis what units?

    14. Likelihood 14 The Y axis is the first derivative of the cumulative probability distribution

    15. Likelihood 15 The log normal distribution

    16. Likelihood 16 Key notes re lognormal distribution Since x is a constant, when calculating likelihoods we often drop the 1/x term If s.d. is fixed, then the entire first term is a constant (also true in the normal) and can be ignored expected value of lognormal is not the mean

    17. Likelihood 17 The beta distribution

    18. Likelihood 18 Shapes of the beta

    19. Likelihood 19 Summary by nature of trials and observations

    20. Likelihood 20

    21. Likelihood 21 Moving from probability distributions to likelihood

    22. Likelihood 22 Probability

    23. Likelihood 23

    24. Likelihood 24

    25. Likelihood 25

    26. Likelihood 26 Rescale to max=1

    27. Likelihood 27 Log likelihoods

    28. Likelihood 28 Multiple observations If observations are independent then

    29. Likelihood 29 Mark recapture example We tagged 100 fish Went back a few days later (after mixing etc) And recaptured 100 fish 5 were tagged. We use Poisson distribution to explore the likelihood of different population sizes

    30. Likelihood 30 What we need Data is number marked, number recaptured, and tags recaptured % tagged is #marked/population size expected recoveries is %tagged*# recaptured expected recoveries is r of the Poisson

    31. Likelihood 31

    32. Likelihood 32

    33. Likelihood 33 Multiple observations Assume we go out twice more, capture 100 animals each time, and 3 and then 4 are captured

    34. Likelihood 34

    35. Likelihood 35 Combining all data

    36. Likelihood 36 The likelihood profile Fix the parameter of interest at discrete values and find the maximum likelihood by searching over all other parameters In the bad old days when people reported confidence intervals, you can use the likelihood profile to calculate a confidence interval add demo from logistic model using macro

    37. Likelihood 37 The concept of support Edwards 1972, “Likelihood” Think of the relative likelihood as the amount of support the data offer for the hypothesis

    38. Likelihood 38 The lognormal distribution

    39. Likelihood 39 Readings on robustness and contradictory data

    40. Likelihood 40 Robustness In the real world, assumptions are not always met For instance, data may be mis-recorded, the wrong animal may be measured, the instrument may have failed, or some major assumption may have been wrong Outliers exist

    41. Likelihood 41

    42. Likelihood 42 What is c?

    43. Likelihood 43 Contaminated data

    44. Likelihood 44 Fit with robust estimation

    45. Likelihood 45 Demonstrate robustness in excel likelihood lecture workbook.xls

    46. Likelihood 46 Contradictory data We often have two independent measures of something, that disagree The problem here is not that an individual data point is contaminated, but that the data set isn’t measuring what we hope

    47. Likelihood 47 The infamous northern cod

    48. Likelihood 48 What they say about r

    49. Likelihood 49 Likelihoods for contradictory data

    50. Likelihood 50 Combined likelihood

    51. Likelihood 51 Challenges in likelihood All probability statements are based on the assumptions of the models We normally do not admit that either data are contaminated, or data sets are not reflecting what we think they are Thus we almost certainly overestimate the confidence in our analysis

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