lec23_11032006 - 10.34 Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #23: Models vs. Data 2: Bayesian view. Models vs. Data Theorem 83: As N expt Æ , the distribution of data, <Y data > Nexpts Æ Normal Î Gaussian 1) Assume Central Limit Theorem (Theorem 83) 2) We assume we have the true model (always wrong) a) We assume we have the true model parameters, or at least the best possible fit θ 3) We assume we know the uncertainties in data σ mean , σ (<Y data >) P(Y data ) = const = > < exp 1 2 mod exp N i i el i data i Y Y σ Since we have the probability density, need to integrate over some Y = const exp(- χ 2 ). χ 2 collapses P(Y ) to 1D χ 2 = ts i N i i el i data i N D S Y Y exp 1 2 mod . . exp = = S.D. is the standard deviation of N expts at condition i. N data P( χ 2 ) Figure 1. Chi-squared distribution. Linear (in parameters) Models Y model = M (x θ to find best fit θ min θ χ 2 ( θ ) min||Y data – M · θ || 2 such that θ {possible}
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This note was uploaded on 11/27/2011 for the course CHEMICAL E 10.302 taught by Professor Clarkcolton during the Fall '04 term at MIT.

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lec23_11032006 - 10.34 Numerical Methods Applied to...

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