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Bayesian inferencePrimer
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LikelihoodPosterior
Prior
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⇡(x) ⇡(y|x)⇡(x|y)
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⇡(x) ⇡(y|x)
⇡(x|y)
Bayes’ theorem
posterior =prior⇥ likelihood
evidence
priorposterior likelihood
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Bayes’ theorem
Parameter identification: Bayesian approach
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Bayes’ theorem
Descriptive formula
Parameter identification: Bayesian approach
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A discrete example of Bayes’ theorem
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This is our prior information for the probability of each face: 1/6
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P=1/6
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Assume that after throwing the dice, you see the above evidence
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Goal: determine the probability of this evidence for each face of the dice
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a
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b
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c
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d
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a b
cd
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a b
cd
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a b
cd
One would never see a dot at the star positions for this face The probability of the evidence is zero
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a b
cd
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Two possibilities (a,c) and (b,d)
a b
cd
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Also two possibilities (a,c) and (b,d)
a b
cd
a b
cd
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a b
cd
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a b
cd
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a
bc
d
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a
bc
d
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ab
c d
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a b
cd
a
bc
d
ab
c d
a
bc
d
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a b
cd
a
bc
d
ab
c d
a
bc
d
Four possibilities
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Four possibilities
a b
cd
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Four possibilities
a b
cd
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a b
cda b
cd
a b
cd
a b
cd
a b
cd
a b
cd
0
2 2
4 4 4
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a b
cda b
cd
a b
cd
a b
cd
a b
cd
a b
cd
0
2 2
4 4 4
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Probability that this was the face of the dice knowing
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P=0
P=⅛
P=¼
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P=0
P=⅛
P=¼
P=1/6
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 201811/1/201545
Stress-strain data
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�
✏
Identify the parameters
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Model
observations=f(parameters, error)
Construct the likelihood function
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Additive noise model
Noise model
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Likelihood function for additive model
Likelihood function
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201550
Constitutive model
Observed data
Constitutive law: linear elasticity
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201551
Prior information on Young’s modulus
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201552
Error model (noise)
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201553
Likelihood function
Likelihood function
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Bayes’ theorem: calculate the posterior
posterior =prior⇥ likelihood
evidence
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⇡(x)
Bayes’ theorem: calculate the posterior
posterior =prior⇥ likelihood
evidence
prior
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⇡(x) ⇡(y|x)
Bayes’ theorem: calculate the posterior
posterior =prior⇥ likelihood
evidence
prior⇡(y|x) = N(y � x✏, 0.0001)⇡(y|x) = ⇡(!) = ⇡(y � f(x))
likelihood
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⇡(x)
⇡(x|y)
Bayes’ theorem: calculate the posterior
posterior =prior⇥ likelihood
evidence
prior
posterior
⇡(y|x) = N(y � x✏, 0.0001)⇡(y|x) = ⇡(!) = ⇡(y � f(x))
⇡(y|x)likelihood
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201558
Posterior probability
⇡(x|y) = ⇡(x)⇡(y|x)R⌦ ⇡(x)⇡(y|x)dx
⇡(x)
⇡(y|x)
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201559
The 99.73% rule: observations
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201560
Propagation of the uncertainty to the constitutive model
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 201811/1/201561
Perfect plasticity
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 201811/1/201562
Perfect plasticity
Constitutive model
x(2)
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201563
Observed data
Modified form for constitutive model
Perfect plasticity
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201564
Perfect plasticity: contour plot of prior in parameter space
parameter 1
parameter 2
“wider” priorfor yield stress-> more infoneeded
“narrow” prior for E
�y0
E
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201565
Posterior probability
Perfect plasticity
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201566
Posterior probability
Perfect plasticity
likelihood for each observation1/�2
(x� µ)2
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201567
Posterior probability
Perfect plasticity
likelihood for each observation
stress modelstress measurement
1/�2
(x� µ)2
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201568
Posterior probability
Perfect plasticity
likelihood for each observation
stress modelstress measurement
1/�2
f(x|µ,�) = 1
�p2⇡
e�(x�µ)2
2�2
(x� µ)2
Difficult to compute the evidence probability: use MCMC
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201569
Markov-Chain Monte Carlo (MCMC) method: parameter space
too small adequate
too large too large
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 20188/31/201570
Perfect plasticity: amplitude plot
