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Modeling Data

Modeling Data. Greg Beckham. Bayes. Fitting Procedure should provide Parameters Error estimates on the parameters A statistical measure of goodness of fit Without a statistical measure of goodness of fit, the model is most likely useless. Bayes ’ Theorem.

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Modeling Data

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  1. Modeling Data Greg Beckham

  2. Bayes • Fitting Procedure should provide • Parameters • Error estimates on the parameters • A statistical measure of goodness of fit • Without a statistical measure of goodness of fit, the model is most likely useless

  3. Bayes’ Theorem • Bayes’ Theorem relates the conditional probability of two events • P(A|B) = P(A) * (P(B|A)/P(B)) • P(A|B) is the probability of A given B • P(A|B) = (P(A) ^ P(B))/P(B) • 0 <= P(A|B) <= 1 • P(A) and P(B) are unconditional probabilities

  4. Example • Suppose population of 60% boys and 40% girls • Girls wear pants and skirts with equal probability, boys always wear pants • If a student is wearing pants (B), what is the probability the student is a girl (A)? • P(B|A) = .5 • P(A) = .4 • P(B) = .5 * .4 + 1 * .6 = .8 • P(A|B) = P(B|A)P(A)/P(B) = (.5 * .4)/.8 = .25

  5. Bayes’ Theorem • Defs • H, some hypothesis • I, collective background data • Can assign some plausibility to H even before we have explicit data • P(H|I) • Bayesian Prior

  6. Bayes’ Theorem • When data is added • When more data is added • Using the chain rule Shows you get the same answer if the data came together

  7. Questions

  8. Linear Least Squares • Fitting N data points (xi, yi), t = 0, …, N-1 to a model that has M adjustable parameters • Functional relationship between measured independent and dependant variables • Minimize of a0…aM-1

  9. Linear Least Squares • How to derive linear least squares • Start by asking Given a particular set of parameters, what is the probability that the data set should have occurred • Plus or minus some fixed Δy • Each data point yi has a measurement error that is independently random and distributed as a normal distribution around true model y(x) • Standard deviations of these are σ

  10. Linear Least Squares • Probability of the data set is the product of the probabilities at each point 15.1.2

  11. Least Squares • Notice the Δy in each term • So the goal is to minimize the error • Where P(model) = P(a0…aM-1) is our prior probability distribution on al models

  12. Least Squares • Take logarithm of 15.1.2 to get • Since N, Δy, σ are constants, minimizing this is equivalent to minimizing 15.1.2

  13. Non-Linear Fit

  14. Non-linear Fit • Fitting depends non-linearly on the set of unknown parameters ak, k = 0,1,…,M-1 • Define a merit function χ2 (chi-squared) and determine best-fit parameters by minimizing the function • Minimization must be done iteratively until it stops decreasing (only changes by some small amount)

  15. Non-linear Fit • Where d is an M-vector and D is an MxM matrix • D is the second derivative of χ2 at any a. • If we are close then • If it is a poor approximation then step down the gradient

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