Post on 08-Apr-2022
11
⇡(x) ⇡(y|x)
⇡(x|y)
Bayes’ theorem
posterior =prior⇥ likelihood
evidence
priorposterior likelihood
8/31/2015 26
a b
cd
One would never see a dot at the star positions for this face The probability of the evidence is zero
Bayesian Inference Primer - Hussein Rappel, Lars Beex, Jack Hale, Stéphane Bordas 201811/1/201545
Stress-strain data
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
56
⇡(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
57
⇡(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