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 Nexpt Æ ∞, the distribution of data, <Ydata>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 σmean, σ(<Ydata>)
⎛ ⎛<Y
i data > −Yi mod el
P(Ydata) = const ∏ exp⎜ − ⎜⎜
⎜ ⎝
σi
i =1
⎝
N exp
⎞
⎟⎟
⎠
2
⎞
⎟
⎟
⎠
Since we have the probability density, need to integrate over some ∆Y = const exp(-χ2).
χ2 collapses P(Y) to 1D
⎛ Yi data − Yi mod el
χ = ∑ ⎜⎜
σi
i =1 ⎝
N exp
2
⎞
⎟⎟
⎠
2
σ i = S .D.
N exp ts
S.D. is the standard deviation of Nexpts at condition i.
P(χ2)
Ndata
Figure 1. Chi-squared distribution.
Linear (in parameters) Models
Ymodel = M(x)·θ
to find best fit θ
minθ χ2(θ)
min||Ydata – M·θ||2
such that θ ∈ {possible}
where (Ydata )i
≡
Yi
σi
M ij
∑σ
j
i
θ i =(M θ )i
∂χ 2
T
= 0 = 2(Y − M θ ) M j
∂θ j
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall
2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD
Month YYYY].
⎛ Yi − ∑ M ijθ j
⎜
j
2
χ = ∑⎜
σi
i ⎜
⎝
0=
1
∂X 2
=∑ 2
∂θ n
i σ
⎞
⎟
⎟
⎟
⎠
2
⎞
⎛
2⎜⎜ Yi − ∑ M ijθ j ⎟⎟(− M cn )
j
⎠
⎝
10.34, Numerical Methods Applied to Chemical Engineering Lecture 23
Prof. William Green
Page 2 of 2
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall
2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD
Month YYYY].