model and parameter identification through bayesian
play

Model and parameter identification through Bayesian inference in - PowerPoint PPT Presentation

Jul 19, 2023 •9 likes •789 views

Model and parameter identification through Bayesian inference in solid mechanics Hussein Rappel h.rappel@gmail.com September 07, 2018 Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 1 / 69 Introduction:

  1. Model and parameter identification through Bayesian inference in solid mechanics Hussein Rappel h.rappel@gmail.com September 07, 2018 Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 1 / 69

  2. Introduction: Probabilistic modelling y 4 3 y = -ax + b 2 a 1 1 x 1 2 3 4 Introduction to Gaussian Processes, Neil Lawrence Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 2 / 69

  3. Introduction: Probabilistic modelling y 4 3 2 1 x 1 2 3 4 Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 3 / 69

  4. Introduction: Probabilistic modelling y 4 3 2 1 x 1 2 3 4 Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 4 / 69

  5. Introduction: Probabilistic modelling y 4 3 2 1 x 1 2 3 4 Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 5 / 69

  6. Introduction: Probabilistic modelling Each point can be written as the model+ a corruption: y 1 = ax + c + ω 1 y 2 = ax + c + ω 2 y 3 = ax + c + ω 3 Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 6 / 69

  7. Introduction: Probabilistic modelling The corruption term can be presented with a probability distribution Pierre-Simon Laplace 1749-1827 (source wikipedia) Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 7 / 69

  8. Introduction: Probabilistic modelling y 4 How can we fit the y = ax + b line, having 3 only one point? 2 1 x 1 2 3 4 Introduction to Gaussian Processes, Neil Lawrence Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 8 / 69

  9. Introduction: Probabilistic modelling y 4 If b is fixed ⇒ a = y-b 3 = x 2 1 x 1 2 3 4 Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 9 / 69

  10. Introduction: Probabilistic modelling y 4 3 2 1 x 1 2 3 4 Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 10 / 69

  11. Introduction: Probabilistic modelling y 4 b ∼ π 1 = ⇒ a ∼ π 2 3 2 1 x 1 2 3 4 Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 11 / 69

  12. Introduction: Probabilistic modelling This is called Bayesian treatment. The model parameters are treated as random variables. Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 12 / 69

  13. Introduction: Bayesian perspective Original belief New belief Observations Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 13 / 69

  14. Introduction: Bayesian formula (inverse probability) likelihood prior posterior � �� � � �� � � �� � π (x) × π (y|x) π (x|y) = π (y) � �� � 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 Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 14 / 69

  15. Introduction: Bayesian formula (inverse probability) π (x|y) ∝ π (x) × π (y|x) Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 15 / 69

  16. BI in computational mechanics σ ε Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 16 / 69

  17. Linear elasticity σ σ = E ε E 1 ε Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 17 / 69

  18. Linear elasticity y = E ε + ω Ω ∼ π ω ( ω ) Capital letters denote a random variable Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 18 / 69

  19. Linear elasticity σ ( ) - ω 2 1 π ω ( ω ) = 2 π s ω exp √ 2 s 2 ω ε Noise PDF is modelled through calibration test. Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 19 / 69

  20. Linear elasticity Bayes’ formula: π (E|y) = π (E) π (y|E) = π (E) π (y|E) π (y) k π (E|y) ∝ π (E) π (y|E) Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 20 / 69

  21. Linear elasticity y = E ε + ω Ω ∼ N( 0 , s 2 ω ) Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 21 / 69

  22. Linear elasticity - (y - E ε ) 2 1 ( ) π (y|E) = √ exp 2 s 2 2 π s ω ω Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 22 / 69

  23. Linear elasticity Posterior: ( ) ( ) - (E-E) 2 - (y-E ε ) 2 π (E|y) ∝ exp exp 2 s 2 2 s 2 E ω Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 23 / 69

  24. Linear elasticity Prediction interval: An estimate of an interval in which an observation will fall, with a certain probability. Credible region: A region of a distribution in which it is believed that a random variable lie with a certain probability. Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 24 / 69

  25. Linear elasticity Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 25 / 69

  26. Prior effect Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 26 / 69

  27. Model uncertainty and input error y = f( x , ε ) + ω Ω ∼ π ω ( ω ) ε is the input variable and x is the parameter vector Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 27 / 69

  28. Model uncertainty and input error Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 28 / 69

  29. Model uncertainty and input error Kennedy-O’Hagan (KOH) framework: y = f( x , ε ) + d( x d , ε ) + ω � �� � y true ε is the input variable and x is the parameter vector Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 29 / 69

  30. Model uncertainty and input error Constant number d 0 Deterministic function ∑ l i= 0 a i ε i Random variable from normal distribution d ∼ N(m, s 2 d ) Random variable from a normal distribution with input dependent mean and variance d ∼ N(m( ε ), s 2 d ( ε )) Gaussian process (GP) ... Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 30 / 69

  31. Model uncertainty and input error Bayes’ formula: π ( x , x d |y) ∝ π ( x ) π ( x d ) π (y| x , x d ) Both material and model error parameters must be inferred. If d( x d , ε ) is deterministic (for simplicity): π ( x , x d |y) ∝ π ( x ) π ( x d ) π ω (y - f( x , ε ) - d( x d , ε )) x d parameter vector of model uncertainty Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 31 / 69

  32. Model uncertainty and input error Bayes’ formula: π ( x , x d |y) ∝ π ( x ) π ( x d ) π (y| x , x d ) Both material and model error parameters must be inferred. If d( x d , ε ) is deterministic (for simplicity): π ( x , x d |y) ∝ π ( x ) π ( x d ) π ω (y - f( x , ε ) - d( x d , ε )) x d parameter vector of model uncertainty Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 31 / 69

  33. Model uncertainty and input error But what about the input error? Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 32 / 69

  34. Model uncertainty and input error y = f( x , ε ) + d( x d , ε ) + ω ε ∗ = ε + ω ε Ω ∼ π ( ω ) Ω ε ∼ π ( ω ε ) Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 33 / 69

  35. Model uncertainty and input error Bayes’ formula: π ( x , x d , ε |y, ε ∗ ) ∝ π (y| x , x d , ε ) π ( ε | ε ∗ ) π ( x ) π ( x d ) ∫ b π ( x , x d |y, ε ∗ ) ∝ 0 π (y| x , x d , ε ) π ( ε | ε ∗ )d ε π ( x ) π ( x d ) π (y| x , x d , ε ) = π ω (y - f( x , ε ) - d( x d , ε )) π ( ε | ε ∗ ) = π ω ε ( ε ∗ - ε ) Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 34 / 69

  36. Model uncertainty and input error Bayes’ formula: π ( x , x d , ε |y, ε ∗ ) ∝ π (y| x , x d , ε ) π ( ε | ε ∗ ) π ( x ) π ( x d ) ∫ b π ( x , x d |y, ε ∗ ) ∝ 0 π (y| x , x d , ε ) π ( ε | ε ∗ )d ε π ( x ) π ( x d ) π (y| x , x d , ε ) = π ω (y - f( x , ε ) - d( x d , ε )) π ( ε | ε ∗ ) = π ω ε ( ε ∗ - ε ) Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 34 / 69

  37. Model uncertainty and input error: Linear elasticity Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 35 / 69

  38. Model uncertainty and input error: Effect of model uncertainty Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 36 / 69

  39. Model uncertainty and input error: Effect of model uncertainty as well as input error Normal distribution with constant parameters Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 37 / 69

  40. Model uncertainty and input error: Effect of model uncertainty as well as input error Normal distribution with an input-dependent mean Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 38 / 69

  41. Model uncertainty and input error: Effect of model uncertainty as well as input error Gaussian process Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 39 / 69

Recommend

I 4 - Bayesian parameter estimation in a normal model STAT 587 (Engineering) Iowa State

I 4 - Bayesian parameter estimation in a normal model STAT 587 (Engineering) Iowa State

I 4 - Bayesian parameter estimation in a normal model STAT 587 (Engineering) Iowa State University September 18, 2020 Bayesian parameter estimation Bayesian parameter estimation Recall that Bayesian parameter estimation involves p ( | y ) =

439 views • 20 slides
On parameter identification in linear stochastic differential equations by Gaussian statistics

On parameter identification in linear stochastic differential equations by Gaussian statistics

Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work On parameter identification in linear stochastic differential equations by Gaussian statistics Shuai Lu

659 views • 36 slides
Bayesian Methods for Parameter Estimation Bayesian vs Frequentist Inference Frequentist Chris

Bayesian Methods for Parameter Estimation Bayesian vs Frequentist Inference Frequentist Chris

Bayesian Methods for Parameter Estimation Bayesian vs Frequentist Inference Frequentist Chris Williams, Division of Informatics Assumes that there is an unknown but fixed parameter University of Edinburgh Estimates with some

208 views • 5 slides
Outline Intro to RL and Bayesian Learning History of Bayesian RL Model-based Bayesian

Outline Intro to RL and Bayesian Learning History of Bayesian RL Model-based Bayesian

Outline Intro to RL and Bayesian Learning History of Bayesian RL Model-based Bayesian RL Prior knowledge, policy optimization, discussion, Bayesian approaches for other RL variants Model-free Bayesian RL Gaussian

1.23k views • 63 slides
Overview Bayesian Methods for Parameter Estimation Introduction to Bayesian Statistics: Learning

Overview Bayesian Methods for Parameter Estimation Introduction to Bayesian Statistics: Learning

Overview Bayesian Methods for Parameter Estimation Introduction to Bayesian Statistics: Learning a Probability Learning the mean of a Gaussian Chris Williams Readings: Bishop 2.1 (Beta), 2.2 (Dirichlet), 2.3.6 School of Informatics,

177 views • 5 slides
HYPOTHESIS TESTING PART I RECAP & OUTLOOK BAYESIAN PARAMETER ESTIMATION FREQUENTIST

HYPOTHESIS TESTING PART I RECAP & OUTLOOK BAYESIAN PARAMETER ESTIMATION FREQUENTIST

INTRODUCTION TO DATA ANALYSIS HYPOTHESIS TESTING PART I RECAP & OUTLOOK BAYESIAN PARAMETER ESTIMATION FREQUENTIST HYPOTHESIS TESTING model captures prior beliefs model captures a hypothetically M M about data-generating process

646 views • 27 slides
Maximum-likelihood and Bayesian parameter estimation Andrea Passerini passerini@disi.unitn.it

Maximum-likelihood and Bayesian parameter estimation Andrea Passerini passerini@disi.unitn.it

Maximum-likelihood and Bayesian parameter estimation Andrea Passerini passerini@disi.unitn.it Machine Learning Maximum-likelihood and Bayesian parameter estimation Parameter estimation Setting Data are sampled from a probability distribution

856 views • 52 slides
An MPEC Formulation for Parameter Identification of Complementarity Systems S. Berard J.C.

An MPEC Formulation for Parameter Identification of Complementarity Systems S. Berard J.C.

An MPEC Formulation for Parameter Identification of Complementarity Systems An MPEC Formulation for Parameter Identification of Complementarity Systems S. Berard J.C. Trinkle Department of Computer Science Rensselaer Polytechnic Institute May

500 views • 36 slides
Accurate parameter estimation for Bayesian network classifiers using hierarchical Dirichlet

Accurate parameter estimation for Bayesian network classifiers using hierarchical Dirichlet

Accurate parameter estimation for Bayesian network classifiers using hierarchical Dirichlet processes Fran cois Petitjean, Wray Buntine , Geoff Webb and Nayyar Zaidi Monash University 2018-09-13 1 / 35 Outline Motivation Bayesian

870 views • 69 slides
CHAPTER 6: EMPIRICAL MODEL IDENTIFICATION Outline of the lesson. Experimental design for

CHAPTER 6: EMPIRICAL MODEL IDENTIFICATION Outline of the lesson. Experimental design for

CHAPTER 6: EMPIRICAL MODEL IDENTIFICATION Outline of the lesson. Experimental design for model building Process reaction curve (graphical) Statistical parameter estimation Workshop CHAPTER 6: EMPIRICAL MODELLING We have invested

259 views • 14 slides
Lecture 6. Bayesian estimation Lecture 6. Bayesian estimation 1 (172) 6. Bayesian estimation

Lecture 6. Bayesian estimation Lecture 6. Bayesian estimation 1 (172) 6. Bayesian estimation

5. Lecture 6. Bayesian estimation Lecture 6. Bayesian estimation 1 (172) 6. Bayesian estimation 6.1. The parameter as a random variable The parameter as a random variable So far we have seen the frequentist approach to statistical inference

687 views • 14 slides
Bayesian model averaging Dr. Jarad Niemi STAT 544 - Iowa State University March 9, 2017 Jarad

Bayesian model averaging Dr. Jarad Niemi STAT 544 - Iowa State University March 9, 2017 Jarad

Bayesian model averaging Dr. Jarad Niemi STAT 544 - Iowa State University March 9, 2017 Jarad Niemi (STAT544@ISU) Bayesian model averaging March 9, 2017 1 / 27 Outline Bayesian model averaging BIC model averaging Model search Parameter

385 views • 27 slides
1 Estimation from outbreaks when R 0 < 1 Estimating R 0 : from epidemic data If case data are

1 Estimation from outbreaks when R 0 < 1 Estimating R 0 : from epidemic data If case data are

Outline Parameter estimation, uncertainty, Estimating R 0 model fitting, model selection, Parameter estimation - Likelihood approaches and sensitivity and uncertainty analysis - Bayesian approaches Fitting more complex models Jamie

240 views • 12 slides
Introduction to Bayesian Inference Frank Wood April 6, 2010 Introduction Overview of Topics

Introduction to Bayesian Inference Frank Wood April 6, 2010 Introduction Overview of Topics

Introduction to Bayesian Inference Frank Wood April 6, 2010 Introduction Overview of Topics Bayesian Analysis Single Parameter Model Bayesian Analysis Recipe Bayesian data analysis can be described as a three step process 1. Set up a full

833 views • 31 slides
Bayesian Hierarchical Models for parameter inference with missing

Bayesian Hierarchical Models for parameter inference with missing

FERMILAB-SLIDES-18-079 Bayesian Hierarchical Models for parameter inference with missing data: Supernova cosmology case study. Marisa Cris)na March, University of

348 views • 13 slides
On the Use of Analytical Techniques for Parameter Identification in Radiation and Particle

On the Use of Analytical Techniques for Parameter Identification in Radiation and Particle

On the Use of Analytical Techniques for Parameter Identification in Radiation and Particle Transport Models L. B. Barichello Instituto de Matem atica e Estat stica Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil

880 views • 63 slides
Bayesian Hierarchical Models for parameter inference with missing

Bayesian Hierarchical Models for parameter inference with missing

Bayesian Hierarchical Models for parameter inference with missing data: Supernova cosmology case study. Marisa Cris)na March, University of Pennsylvania New

454 views • 18 slides
Meta-Bayesian Analysis A Bayesian decision-theoretic analysis of Bayesian inference under model

Meta-Bayesian Analysis A Bayesian decision-theoretic analysis of Bayesian inference under model

Meta-Bayesian Analysis A Bayesian decision-theoretic analysis of Bayesian inference under model misspecification Jun Yang joint work with Daniel Roy Department of Statistical Sciences University of Toronto World Congress in Probability and

190 views • 17 slides
Gravitational Wave Data Analysis: II. Model Selection and Parameter Estimation Chris Van Den

Gravitational Wave Data Analysis: II. Model Selection and Parameter Estimation Chris Van Den

Gravitational Wave Data Analysis: II. Model Selection and Parameter Estimation Chris Van Den Broeck Kavli RISE Summer School on Gravitational Waves, Cambridge, UK, 23-27 September 2019 Bayesian inference Aim: use available data to

707 views • 47 slides
A Short Introduction to Bayesian Optimization With applications to parameter tuning on

A Short Introduction to Bayesian Optimization With applications to parameter tuning on

A Short Introduction to Bayesian Optimization With applications to parameter tuning on accelerators Johannes Kirschner 28th February 2018 ICFA Workshop on Machine Learning for Accelerator Control Solve x = arg max f ( x ) x X 0

511 views • 36 slides
RfG 1 Year Implementation : Timeline National Parameter identification Mod 1 Structure /

RfG 1 Year Implementation : Timeline National Parameter identification Mod 1 Structure /

RfG 1 Year Implementation : Timeline National Parameter identification Mod 1 Structure / Banding Mod 2 Compliance Mod 3 General Mod 4 Fault Ride Through Mod 5 Voltage and Reactive Power Mod 6 Frequency 3/11/14 7/09/15

471 views • 5 slides
Single-parameter models: Binomial data Applied Bayesian Statistics Dr. Earvin Balderama

Single-parameter models: Binomial data Applied Bayesian Statistics Dr. Earvin Balderama

Single-parameter models: Binomial data Applied Bayesian Statistics Dr. Earvin Balderama Department of Mathematics & Statistics Loyola University Chicago September 5, 2017 Beta-Binomial model 1 Last edited September 5, 2017 by Earvin

292 views • 13 slides
Single-parameter models: Gaussian (normal) data Applied Bayesian Statistics Dr. Earvin Balderama

Single-parameter models: Gaussian (normal) data Applied Bayesian Statistics Dr. Earvin Balderama

Single-parameter models: Gaussian (normal) data Applied Bayesian Statistics Dr. Earvin Balderama Department of Mathematics & Statistics Loyola University Chicago September 19, 2017 The Gaussian (normal) model 1 Last edited September 8,

251 views • 12 slides
Single-parameter models: Poisson data Applied Bayesian Statistics Dr. Earvin Balderama

Single-parameter models: Poisson data Applied Bayesian Statistics Dr. Earvin Balderama

Single-parameter models: Poisson data Applied Bayesian Statistics Dr. Earvin Balderama Department of Mathematics & Statistics Loyola University Chicago September 14, 2017 Poisson-Gamma model 1 Last edited September 8, 2017 by Earvin

321 views • 11 slides

More recommend