lec27_11172006 - 10.34, Numerical Methods Applied to...

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Professor William H. Green Lecture #27: Models vs. Data Recapitulation. Monte Carlo. Least Squares Fit Y model = Σθ i f i (x ) θ best θ 2 θ 1 χ 2 min χ 2 = const Use probability ( χ 2 > χ 2 measured ) > tol to set contour boundary for consistent or not consistent. Nonlinear Least Squares Away from “ θ best ”, no idea about contours. Close to minima, looks like ellipses. Nonlinear case can have numerous local minima and arbitrary shape. It is possible for the problem to be poorly constrained, to have multiple minima, and to have bad directions. The result is big error bars and a complete mess. Y model ( θ , x ) Y model ( θ best ) + Y model / θ ( θ - θ best ) + O(( ∆θ ) 2 ) c l o s e t o θ best will be linear neglect can have more minima don t know parameter values well in this direction θ 2 θ 1 Standard confidence intervals – covariance matrix, assume ellipses for confidence intervals. To tell the actual shape of the region, use Bayesian view: report probability distribution of
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lec27_11172006 - 10.34, Numerical Methods Applied to...

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