Statistical Estimation of Hyperparameters a s are determined

  • Slides: 50
Download presentation

Statistical Estimation of Hyperparameters a, s are determined so as to maximize the marginal

Statistical Estimation of Hyperparameters a, s are determined so as to maximize the marginal likelihood Pr{Y=y|a, s} with respect to a, s. EM (Expectation Maximization) Algorithm Original Image Degraded Image Marginalized with respect to X Marginal Likelihood 10 November, 2011 次世代情報処理技術とその応用＠早稲田 大学研究開発センター 15

Gaussian Graphical Model (Gauss Markov Random Fields) Multidimensional Gauss Integral Formulas Maximum Likelihood Estimation

Gaussian Graphical Model (Gauss Markov Random Fields) Multidimensional Gauss Integral Formulas Maximum Likelihood Estimation EM Algorithm 10 November, 2011 次世代情報処理技術とその応用＠早稲田 大学研究開発センター 17

1次元信号に対する例 Original Signal 200 EM Algorithm 100 0 Degraded Signal 200 0 127 255

1次元信号に対する例 Original Signal 200 EM Algorithm 100 0 Degraded Signal 200 0 127 255 100 0 Estimated Signal 200 100 0 10 November, 2011 次世代情報処理技術とその応用＠早稲田 大学研究開発センター 18

Bayesian Image Analysis by Gaussian Graphical Model Iteration Procedure of EM algorithm in Gaussian

Bayesian Image Analysis by Gaussian Graphical Model Iteration Procedure of EM algorithm in Gaussian Graphical Model EM 10 November, 2011 次世代情報処理技術とその応用＠早稲田 大学研究開発センター 19

Image Restoration by Gaussian Graphical Model and Conventional Filters Original Image Degraded Image (s=40)

Image Restoration by Gaussian Graphical Model and Conventional Filters Original Image Degraded Image (s=40) MSE Gaussian Graphical Model Lowpass Filter Median Filter 315 (3 x 3) 388 (5 x 5) 413 (3 x 3) 486 (5 x 5) 445 V: Set of all the pixels Gaussian Graphical Model 10 November, 2011 (3 x 3) Lowpass (5 x 5) Median 次世代情報処理技術とその応用＠早稲田 大学研究開発センター 20

Fixed Point Equation and Iterative Method Fixed Point Equation Iterative Method 10 November, 2011

Fixed Point Equation and Iterative Method Fixed Point Equation Iterative Method 10 November, 2011 次世代情報処理技術とその応用＠早稲田 大学研究開発センター 34

Maximization of Marginal Likelihood by EM Algorithm Exact 10 November, 2011 次世代情報処理技術とその応用＠早稲田 大学研究開発センター Loopy

Maximization of Marginal Likelihood by EM Algorithm Exact 10 November, 2011 次世代情報処理技術とその応用＠早稲田 大学研究開発センター Loopy Belief Propagation 37

Image Restoration by Gaussian Graphical Model Original Image Degraded Image Belief Propagation Exact MSE:

Image Restoration by Gaussian Graphical Model Original Image Degraded Image Belief Propagation Exact MSE: 325 MSE: 315 MSE: 1512 Lowpass Filter MSE: 411 10 November, 2011 Wiener Filter Median Filter MSE: 545 MSE: 447 次世代情報処理技術とその応用＠早稲田 大学研究開発センター 38

Markov Random Field Output Input Digital Images Inpainting based on MRF M. Yasuda, J.

Markov Random Field Output Input Digital Images Inpainting based on MRF M. Yasuda, J. Ohkubo and K. Tanaka: Proceedings of CIMCA&IAWTIC 2005. 10 November, 2011 次世代情報処理技術とその応用＠早稲田 大学研究開発センター 39

Statistical Performance Estimation in Image Restoration for Bayesian Image Analysis Mean Square Error Spin

Statistical Performance Estimation in Image Restoration for Bayesian Image Analysis Mean Square Error Spin Glass Theory in Statistical Mechanics s=40 s=1 a 10 November, 2011 次世代情報処理技術とその応用＠早稲田 大学研究開発センター a 42

Belief Propagation for Bayesian Networks 10 November, 2011 次世代情報処理技術とその応用＠早稲田 大学研究開発センター 44

Belief Propagation for Bayesian Networks 10 November, 2011 次世代情報処理技術とその応用＠早稲田 大学研究開発センター 44

Factor Graph Representations of Bayesian Networks and Belief Propagations 10 November, 2011 次世代情報処理技術とその応用＠早稲田 大学研究開発センター

Factor Graph Representations of Bayesian Networks and Belief Propagations 10 November, 2011 次世代情報処理技術とその応用＠早稲田 大学研究開発センター 45

Error Correcting Codes and Belief Propagation Fundamental Concept for Turbo Codes and LDPC Codes

Error Correcting Codes and Belief Propagation Fundamental Concept for Turbo Codes and LDPC Codes 10 November, 2011 次世代情報処理技術とその応用＠早稲田 大学研究開発センター 46

Hyperparameters estimation for Bayesian positive system identification viaHyperparameters estimation for Bayesian positive system identification via
Hyperparameters biasvariance tradeoff validation Supervised learning outline revisitedHyperparameters biasvariance tradeoff validation Supervised learning outline revisited
Stealing Hyperparameters in Machine Learning Binghui Wang andStealing Hyperparameters in Machine Learning Binghui Wang and
Automatically Determining Hyperparameters Source of iamge https mediumAutomatically Determining Hyperparameters Source of iamge https medium
Automatically Determining Hyperparameters Source of iamge https mediumAutomatically Determining Hyperparameters Source of iamge https medium
Ridge and Lasso Hyperparameters CS 109 A IntroductionRidge and Lasso Hyperparameters CS 109 A Introduction
Fast Bayesian Optimization of Machine Learning Hyperparameters onFast Bayesian Optimization of Machine Learning Hyperparameters on
Point estimation interval estimation Point estimation Desirable propertiesPoint estimation interval estimation Point estimation Desirable properties
Estimation Making Educated Guesses Point Estimation Interval EstimationEstimation Making Educated Guesses Point Estimation Interval Estimation
Software Project Estimation Estimation Project Estimation determines howSoftware Project Estimation Estimation Project Estimation determines how
Estimation Point Estimation Statistical Inference for Managers LectureEstimation Point Estimation Statistical Inference for Managers Lecture
Chapter 4 Statistical Inferences Estimation Chapter 4 StatisticalChapter 4 Statistical Inferences Estimation Chapter 4 Statistical
Statistical models estimation and confidence intervals Statistical modelsStatistical models estimation and confidence intervals Statistical models
Biostatistics statistical software IV Statistical estimation confidence intervalsBiostatistics statistical software IV Statistical estimation confidence intervals
STATISTICAL INFERENCE PART I POINT ESTIMATION 1 STATISTICALSTATISTICAL INFERENCE PART I POINT ESTIMATION 1 STATISTICAL
STATISTICAL INFERENCE PART I POINT ESTIMATION 1 STATISTICALSTATISTICAL INFERENCE PART I POINT ESTIMATION 1 STATISTICAL
Statistical Testing Statistical Testing n n n StatisticalStatistical Testing Statistical Testing n n n Statistical
Chapter 7 Statistical Inference Statistical Inference Statistical inferenceChapter 7 Statistical Inference Statistical Inference Statistical inference
CHAPTER 9 ESTIMATION Outline Estimation Point estimator IntervalCHAPTER 9 ESTIMATION Outline Estimation Point estimator Interval
Estimation Econometra ADE Estimation We assume we haveEstimation Econometra ADE Estimation We assume we have
Estimation in Marginal Models GEE and Robust EstimationEstimation in Marginal Models GEE and Robust Estimation
Chapter 4 Demand Estimation The estimation of aChapter 4 Demand Estimation The estimation of a
Estimation CONFIDENTIAL COPYRIGHT 2018 Estimation Course Skills toEstimation CONFIDENTIAL COPYRIGHT 2018 Estimation Course Skills to
Presentation Topic 1 COCOMO ESTIMATION 2 Cocomo estimationPresentation Topic 1 COCOMO ESTIMATION 2 Cocomo estimation