Jensen’s Inequality and Bayes Theorem

1. Jensen’s Inequality and Bayes Theorem Seminar 2 Likelihood Methods in Forest Ecology October 9th – 20th , 2006

2. Functions and limits If we have a function f(x) and a number a in the domain of f, then the limit of f(x) is the value that the function approaches as x gets very close to a. Calculus slides from Hobbs..,.

3. The derivative

4. The derivative The derivative is the instantaneous rate of change in a function at a point. It can be very useful to think of this as a slope of a line tangent to that single point.

10. Taylor series

12. Recall that…

14. Taylor series one variable

15. Taylor series two variables

16. For most functions we can approximate a complex function by a simple function by using the first couple of terms of a Taylor expansion

17. Jensen’s inequality &Bayes Theorem

18. Variance & Jensen’s inequality Given a non-linear function of x, f(x) and a set of x values with a mean of

19. Jensen’s inequality

20. Delta Method • Provides a measure of the difference between and by using T.S. • Difference is proportional to curvature (f’’(x)) and to scatter (Variance)

21. An example: Quantifying the effect of DD, DI and supply on reef fishes A=adult fishes S=supply of recruits a,b=fitted parameters Schmitt et al. 1999

22. Schmitt et al. 1999

23. b-> S-> a=1 Schmitt et al. 1999

24. Ls Ldi Ldd Switch between DD and DI at S=a/b=14.1 settlers/sq-m

25. What is the effect of removing dd? • Mean natural settler density = 24.5 settlers/sq m • Therefore, removing DD could increase fish density almost threefold? No! Where does the system lie under natural conditions?

26. But….Jensen’s inequality

27. Some rules of probability assuming independence A B

28. Bayes Theorem

29. Bayes Theorem

30. Bayes Theorem ?

31. For a set of mutually exclusive hypotheses…..

32. Bolker

33. An example from medical testing

34. An example from medical testing

35. ill Test + Not ill

36. Bayes Theorem Rarely known Hard to integrate function MCMC methods