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Dive into calculus slides discussing functions and limits, derivatives, Taylor series, Jensen's inequality, and Bayes Theorem in forest ecology. Understand variance and the Delta Method. Utilize an example to quantify effects on reef fishes.
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Jensen’s Inequality and Bayes Theorem Seminar 2 Likelihood Methods in Forest Ecology October 9th – 20th , 2006
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..,.
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.
For most functions we can approximate a complex function by a simple function by using the first couple of terms of a Taylor expansion
Variance & Jensen’s inequality Given a non-linear function of x, f(x) and a set of x values with a mean of
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)
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
b-> S-> a=1 Schmitt et al. 1999
Ls Ldi Ldd Switch between DD and DI at S=a/b=14.1 settlers/sq-m
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?
Some rules of probability assuming independence A B
ill Test + Not ill
Bayes Theorem Rarely known Hard to integrate function MCMC methods
