Probabilistic modeling natural way to treat data

(1)

Probabilistic modeling natural way to treat data

Hussein Rappel

University of Luxembourg h.rappel@gmail.com

February 12, 2019

(2)

y

x 1

2 3 4

1 2 3 4

a 1

y = −ax + b

Introduction to Gaussian Processes, Neil Lawrence

2 / 34

(3)

y

x 1

2 3 4

1 2 3 4

(4)

y

x 1

2 3 4

1 2 3 4

4 / 34

(5)

y

x 1

2 3 4

1 2 3 4

(6)

6 / 34

(7)

Each point can be written as themodel+ a corruption:

y₁ = ax + c + ω₁ y₂ = ax + c + ω₂ y₃ = ax + c + ω₃

ω is the difference between real world and model which can be presented by a probability distribution.

We call ω noise!

(8)

What if our observations are less than model parameters?

Underdetermined system

8 / 34

(9)

y

x 1

2 3 4

1 2 3 4

How can we fit the y = ax + b line, having only one point?

Introduction to Gaussian Processes, Neil Lawrence

(10)

y

x 1

2 3 4

1 2 3 4

If b is fixed =⇒ a = ^{y −b}_x

10 / 34

(11)

y

x 1

2 3 4

1 2 3 4

(12)

y

x 1

2 3 4

1 2 3 4

b ∼ π₁ =⇒ a ∼ π₂

12 / 34

(13)

I This is called Bayesian treatment.

I The model parameters are treated asrandom variables.

(14)

Bayesian perspective

Original belief

Observations

New belief

14 / 34

(15)

Bayesian formula (inverse probability)

posterior

z }| { π(x |y ) =

prior

z }| { π(x )×

likelihood

z }| { π(y |x ) π(y )

| {z }

evidence

y :=observation x :=parameter π(x ) :=original belief

π(y |x ) :=given by the mathematical model that relates y to x π(y ) :=is a constant number

(16)

Bayesian formula (inverse probability)

π(x |y ) ∝ π(x ) × π(y |x )

16 / 34

(17)

BI in computational mechanics

σ

(18)

Linear elasticity

σ = E 

σ

E

1

18 / 34

(19)

Linear elasticity

y = E + ω Ω ∼ πω(ω)

Capital letters denote a random variable

(20)

Linear elasticity

σ

π_ω(ω) = ^√¹

2πsωexp−_2s^ω²₂

ω

Noise PDF is modeled through calibration test.

20 / 34

(21)

Linear elasticity

Bayes’ formula:

π(E |y ) = π(E )π(y |E )

π(y ) = π(E )π(y |E ) k

π(E |y ) ∝ π(E )π(y |E )

(22)

Linear elasticity

y = E + ω Ω ∼ N(0, s_ω²)

22 / 34

(23)

Linear elasticity

π(y |E ) = √1 2πsω

exp−(y − E )² 2sω²

(24)

Linear elasticity

Posterior:

π(E |y ) ∝exp− ^{(E −E )}²

2sE²

exp−^{(y −E )}_2s₂ ²

ω

24 / 34

(25)

Linear elasticity

I Prediction interval: An estimate of an interval in which an observation will fall, with a certain probability.

I Credible region:A region of a distribution in which it is believed that a random variable lie with a certain probability.

(26)

Linear elasticity

I Increase in number of observations/measurements makes us more sure of identification result.

26 / 34

(27)

Prior effect

I Increase in number of observations/measurements decreases the effect of prior.

(28)

Conclusion

I Probability is the natural way of dealing with

uncertainties/unknowns (what Laplace calls it our ignorance).

I From Bayesian perspective (inverse probability) the parameters are treated asrandom variables.

I The same logic can be used to model other kinds of

uncertainties/unknowns e.g. model uncertainties and material variability.

I In Bayesian paradigm our assumptions are clearly stated (e.g. the prior, model and ...).

I As the number of observation/measurements increases we become more sure of our identification results.

28 / 34

(29)

Conclusion

(30)

Conclusion

30 / 34

(31)

Conclusion

(32)

Conclusion

32 / 34

(33)

Acknowledgement

The DRIVEN project has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No 811099.

(34)

Some references

I P.S. marquis de Laplace, 1902. A philosophical essay on probabilities.

Translated by F.W. Truscott and F.L. Emory, Wiley.

I E.T. Jaynes, 2003. Probability theory: The logic of science. Cambridge university press.

I D.J. MacKay, 2003. Information theory, inference and learning algorithms.

Cambridge university press.

I A. Gelman et al., 2013. Bayesian data analysis. Chapman and Hall/CRC.

I Courses by N. Lawrence. http://inverseprobability.com/

References by Legatoteam

I H. Rappel et al., 2018. Bayesian inference to identify parameters in viscoelasticity. Mechanics of Time-Dependent Materials.

I H. Rappel et al., 2018. Identifying elastoplastic parameters with Bayes’ theorem considering output error, input error and model uncertainty. Probabilistic Engineering Mechanics.

I H. Rappel et al., 2019. A tutorial on Bayesian inference to identify material parameters in solid mechanics. Archives of Computational Methods in Engineering.

I H. Rappel and L.A.A. Beex, 2019. Estimating fibres’ material parameter distributions from limited data with the help of Bayesian inference. European Journal of Mechanics-A/Solids.

34 / 34