Example: Ground Motion Attenuation

 Madlyn Merritt
 5 months ago
 Views:
Transcription
1 Example: Ground Motion Attenuation Problem: Predict the probability distribution for Peak Ground Acceleration (PGA), the level of ground shaking caused by an earthquake Earthquake records are used to update the predictive probability for PGA based on earthquake magnitude M, distance to the fault d, and local soil conditions G= ( G, G ) (soil classified A, B or C) Wellknown attenuation relation developed by Boore, Joyner and 2 2 Fumal (BJF, 1993). Let R= d + h, h= fictitious depth: 2 ( ) ( ) ( ) ( ) log PGA = b + b M 6 + b M 6 + b R + b log R + b G + b G + ε B 7 qmdgbh (,, ;, ) + ε (where b= ( b1,..., b7)) ε = uncertain prediction error modelled as (0, σ) (max. entropy model) B C C
2 Example: Ground Motion Attenuation Goal: Estimate p( PGA U, D, M) for input U = ( M, d, G) D Available set of data where U = ( M, d, G ) n n n n (magnitude, distance, soil conditions), and corresponding Yn = ( PGA) n are data from earthquakes at various sites (we use =271 ground motion records from 20 earthquakes) Model class M is from BJF model with specified prior PDF over the model parameters θ = (,, bhσ) Robust posterior predictive probability model: (,D, ) = (, ) (, ) M D M = { U, Y : n= 1,..., } ppgauθ (, ) n p PGA U p PGA U θ p θ dθ n
3 Bayesian Updating Bayes Theorem: p θ D, = cp θ M M p Yn Un, θ n= 1 Computing Optimal Posterior Predictive Model ( ) ( ) ( ) Find optimal (most probable values) of parameters which 1 maximize, then to (Laplace s asymptotic approx.): ( D M) p θ, (, DM, ) ppgauθ (, ˆ) ppgau O( ) Assumes M is globally identifiable on ( is unique) and need a large amount of data for an acceptable approximation (updated PDF will then have single sharp peak). Then no need to evaluate and is insensitive to the choice of prior PDF θˆ Parameter estimation (i.e. using ) is reasonable only under these conditions; otherwise, spurious reduction in uncertainty in predictions θˆ D ˆ θ θˆ c
4 Bayesian Updating (Continued) Computing Robust Posterior Predictive Model c ormalizing constant, and p θ,, is difficult to evaluate However, if we can generate M samples from p( θ D ), M, then we can approximate the robust PDF by the corresponding sample mean: (, D, ) = (, ) (, ) M D M p PGA U p PGA U θ p θ dθ 1 M M k= 1 ( ) D M { θ, k = 1,, M } k (, θ ) k ppgau These samples can be obtained using stochastic simulation methods, e.g. Gibbs sampler, MetropolisHastings (more later)
5 Sampling Posterior (Updated) PDF Samples generated using Markov Chain Monte Carlo Prior PDF chosen to reflect knowledge (or lack thereof) For the regression coefficients b, take i.i.d. (0,10), i.e. each has zero mean and standard deviation of 10 (very flat) For the depth parameter h (km), lognormal distribution with mean and variance based on the depth of earthquakes in the data set b i s h
6 MCMC Samples from Posterior PDF Model Class 4 b 1 b 2 b 5 b 6 b 7 h
7 Posterior Samples: Model Class 4 b 2 b 5 b 6 b 7 h b 1 b 2 b 5 x BJF 93 < 1 σ < 2 σ > 2 σ b 6 b 7
8 Optimal vs. Robust Predictive Analysis Results using MCMC samples for robust PDF compared to results for optimal PDF computed from values reported in BJF 93 User input U=(M, d, G) Magnitude 7.0 Distance to fault 30km Site geology is stiff soil
9 Posterior Predictive Analysis 1971 San Fernando M 6.6 JPL Building km 1987 Whittier arrows M Old House Rd 19.0 km 1991 Sierra Madre M S Wilson Ave 18.1 km 1994 orthridge M Sierra Madre Villa 36.2 km
10 Model Class Selection Bayesian model class selection (Beck and Yuen 2004) for set M of candidate models p( D Mi) P( Mi M) P( M D,M) =, i= 1,..., I i I j= 1 p ( D M j) P( M j M) The evidence for model class M i is given by p ( D M ) p( D M, θ ) p( θ M ) dθ = i i i i i i Evaluating the evidence directly by stochastic simulation would require sampling from the prior, which is typically inefficient
11 Model Class Selection (Continued) The evidence may expressed as ( D Mi ) p( D M θ ) p( θ M ) p( θ M D) dθ H p( θ M D) ln p = ln i, i i i i i, i + i i, First term may be approximated from MCMC samples k = 1 ( D M ) ( M ) ( M D) ln p i, θi p θi i p θi i, dθi 1 ln p( D M, ˆ ) ( ˆ ) i θk p θ k Mi umerous methods for approximating information entropy from samples
12 Model Class Selection (Continued) ew result generalizes to any model class the conclusion by Beck and Yuen (2004) made for a globally identifiable model class using an asymptotic approximation of the evidence for the model class: ( D Mi ) ( D M ) ( M ) ( M D) p( θ M D) p( θ M D) dθ ln p = ln p, θ p θ p θ, dθ ln,, i i i i i i i i i i i i ( θ i Mi, D) p ( θ M ) p = ln p ( D Mi, θ i) p( θi Mi, D) dθi ln p( θi Mi, D) dθi i i = Data Fit  Expected Information Gained from Data
13 Model Class Selection ( ) ( ) ( ) ( ) PGA = b1 + b2 M + b3 M + b4 R + b5 10 R + b6gb + b7gc + ε R = d + h log 6 6 log, Model Class b 1 b 2 b 3 b 4 b 5 b 6 b 7 h σ Prob. (%) BJF Model (0.327) (0.034) (0.039) (0.002) (0.252) (0.038) (2.631) Model (0.326) (0.023) (0.002) (0.254) (0.037) (2.534) Model (0.103) (0.034) (0.040) (0.064) (0.038) (1.510) Model (0.108) (0.023) (0.063) (1.162) Model (0.043) (0.026) (0.001) (0.039) (0.041) (1.664) Model (0.114) (0.025) (0.072) (1.720) Model (0.022) (0.021) (0.033) (0.034) (1.041)
14 Comparison by Mean Data Fit Only ( ) ( ) ( ) ( ) PGA = b1 + b2 M + b3 M + b4 R + b5 10 R + b6gb + b7gc + ε R = d + h log 6 6 log, Model Class b 1 b 2 b 3 b 4 b 5 b 6 b 7 h σ Data Fit (%) BJF Model (0.327) (0.034) (0.039) (0.002) (0.252) (0.038) (2.631) Model (0.326) (0.023) (0.002) (0.254) (0.037) (2.534) Model (0.103) (0.034) (0.040) (0.064) (0.038) (1.510) Model (0.108) (0.023) (0.063) (1.162) Model (0.043) (0.026) (0.001) (0.039) (0.041) (1.664) Model (0.114) (0.025) (0.072) (1.720) Model (0.022) (0.021) (0.033) (0.034) (1.041)
15 Posterior Samples : Model Class 7 b 5 b 6 b 7 h b 2 b 5 x BJF 93 < 1 σ < 2 σ > 2 σ b 6 b 7
Risk Estimation and Uncertainty Quantification by Markov Chain Monte Carlo Methods
Risk Estimation and Uncertainty Quantification by Markov Chain Monte Carlo Methods Konstantin Zuev Institute for Risk and Uncertainty University of Liverpool http://www.liv.ac.uk/riskanduncertainty/staff/kzuev/
More informationMarkov Chain Monte Carlo, Numerical Integration
Markov Chain Monte Carlo, Numerical Integration (See Statistics) Trevor Gallen Fall 2015 1 / 1 Agenda Numerical Integration: MCMC methods Estimating Markov Chains Estimating latent variables 2 / 1 Numerical
More informationTutorial on Probabilistic Programming with PyMC3
185.A83 Machine Learning for Health Informatics 2017S, VU, 2.0 h, 3.0 ECTS Tutorial 0204.04.2017 Tutorial on Probabilistic Programming with PyMC3 florian.endel@tuwien.ac.at http://hcikdd.org/machinelearningforhealthinformaticscourse
More informationMetropolisHastings Algorithm
Strength of the Gibbs sampler MetropolisHastings Algorithm Easy algorithm to think about. Exploits the factorization properties of the joint probability distribution. No difficult choices to be made to
More informationApproximate Bayesian Computation: a simulation based approach to inference
Approximate Bayesian Computation: a simulation based approach to inference Richard Wilkinson Simon Tavaré 2 Department of Probability and Statistics University of Sheffield 2 Department of Applied Mathematics
More informationBayesian Inference in Astronomy & Astrophysics A Short Course
Bayesian Inference in Astronomy & Astrophysics A Short Course Tom Loredo Dept. of Astronomy, Cornell University p.1/37 Five Lectures Overview of Bayesian Inference From Gaussians to Periodograms Learning
More informationGeneral Construction of Irreversible Kernel in Markov Chain Monte Carlo
General Construction of Irreversible Kernel in Markov Chain Monte Carlo Metropolis heat bath Suwa Todo Department of Applied Physics, The University of Tokyo Department of Physics, Boston University (from
More informationASYMPTOTICALLY INDEPENDENT MARKOV SAMPLING: A NEW MARKOV CHAIN MONTE CARLO SCHEME FOR BAYESIAN INFERENCE
International Journal for Uncertainty Quantification, 3 (5): 445 474 (213) ASYMPTOTICALLY IDEPEDET MARKOV SAMPLIG: A EW MARKOV CHAI MOTE CARLO SCHEME FOR BAYESIA IFERECE James L. Beck & Konstantin M. Zuev
More informationCPSC 540: Machine Learning
CPSC 540: Machine Learning MCMC and NonParametric Bayes Mark Schmidt University of British Columbia Winter 2016 Admin I went through project proposals: Some of you got a message on Piazza. No news is
More informationForecast combination and model averaging using predictive measures. Jana Eklund and Sune Karlsson Stockholm School of Economics
Forecast combination and model averaging using predictive measures Jana Eklund and Sune Karlsson Stockholm School of Economics 1 Introduction Combining forecasts robustifies and improves on individual
More informationStat 535 C  Statistical Computing & Monte Carlo Methods. Arnaud Doucet.
Stat 535 C  Statistical Computing & Monte Carlo Methods Arnaud Doucet Email: arnaud@cs.ubc.ca 1 1.1 Outline Introduction to Markov chain Monte Carlo The Gibbs Sampler Examples Overview of the Lecture
More informationOn the Optimal Scaling of the Modified MetropolisHastings algorithm
On the Optimal Scaling of the Modified MetropolisHastings algorithm K. M. Zuev & J. L. Beck Division of Engineering and Applied Science California Institute of Technology, MC 444, Pasadena, CA 925, USA
More informationEstimation of Operational Risk Capital Charge under Parameter Uncertainty
Estimation of Operational Risk Capital Charge under Parameter Uncertainty Pavel V. Shevchenko Principal Research Scientist, CSIRO Mathematical and Information Sciences, Sydney, Locked Bag 17, North Ryde,
More informationDensity Estimation. Seungjin Choi
Density Estimation Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongamro, Namgu, Pohang 37673, Korea seungjin@postech.ac.kr http://mlg.postech.ac.kr/
More informationMarkov Networks.
Markov Networks www.biostat.wisc.edu/~dpage/cs760/ Goals for the lecture you should understand the following concepts Markov network syntax Markov network semantics Potential functions Partition function
More informationDSGE Methods. Estimation of DSGE models: Maximum Likelihood & Bayesian. Willi Mutschler, M.Sc.
DSGE Methods Estimation of DSGE models: Maximum Likelihood & Bayesian Willi Mutschler, M.Sc. Institute of Econometrics and Economic Statistics University of Münster willi.mutschler@unimuenster.de Summer
More informationCSC 446 Notes: Lecture 13
CSC 446 Notes: Lecture 3 The Problem We have already studied how to calculate the probability of a variable or variables using the message passing method. However, there are some times when the structure
More informationTools for Parameter Estimation and Propagation of Uncertainty
Tools for Parameter Estimation and Propagation of Uncertainty Brian Borchers Department of Mathematics New Mexico Tech Socorro, NM 87801 borchers@nmt.edu Outline Models, parameters, parameter estimation,
More informationStatistical Machine Learning Lecture 8: Markov Chain Monte Carlo Sampling
1 / 27 Statistical Machine Learning Lecture 8: Markov Chain Monte Carlo Sampling Melih Kandemir Özyeğin University, İstanbul, Turkey 2 / 27 Monte Carlo Integration The big question : Evaluate E p(z) [f(z)]
More informationPetr Volf. Model for Difference of Two Series of Poissonlike Count Data
Petr Volf Institute of Information Theory and Automation Academy of Sciences of the Czech Republic Pod vodárenskou věží 4, 182 8 Praha 8 email: volf@utia.cas.cz Model for Difference of Two Series of Poissonlike
More informationApproximate Inference using MCMC
Approximate Inference using MCMC 9.520 Class 22 Ruslan Salakhutdinov BCS and CSAIL, MIT 1 Plan 1. Introduction/Notation. 2. Examples of successful Bayesian models. 3. Basic Sampling Algorithms. 4. Markov
More informationFitting Complex PK/PD Models with Stan
Fitting Complex PK/PD Models with Stan Michael Betancourt @betanalpha, @mcmc_stan University of Warwick Bayes Pharma Novartis Basel Campus May 20, 2015 Because of the clinical applications, precise PK/PD
More informationCondensed Table of Contents for Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control by J. C.
Condensed Table of Contents for Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control by J. C. Spall John Wiley and Sons, Inc., 2003 Preface... xiii 1. Stochastic Search
More informationLecture 2: From Linear Regression to Kalman Filter and Beyond
Lecture 2: From Linear Regression to Kalman Filter and Beyond January 18, 2017 Contents 1 Batch and Recursive Estimation 2 Towards Bayesian Filtering 3 Kalman Filter and Bayesian Filtering and Smoothing
More informationSAMPLING ALGORITHMS. In general. Inference in Bayesian models
SAMPLING ALGORITHMS SAMPLING ALGORITHMS In general A sampling algorithm is an algorithm that outputs samples x 1, x 2,... from a given distribution P or density p. Sampling algorithms can for example be
More informationBayesian linear regression
Bayesian linear regression Linear regression is the basis of most statistical modeling. The model is Y i = X T i β + ε i, where Y i is the continuous response X i = (X i1,..., X ip ) T is the corresponding
More informationA generalization of the Multipletry Metropolis algorithm for Bayesian estimation and model selection
A generalization of the Multipletry Metropolis algorithm for Bayesian estimation and model selection Silvia Pandolfi Francesco Bartolucci Nial Friel University of Perugia, IT University of Perugia, IT
More informationAPM 541: Stochastic Modelling in Biology Bayesian Inference. Jay Taylor Fall Jay Taylor (ASU) APM 541 Fall / 53
APM 541: Stochastic Modelling in Biology Bayesian Inference Jay Taylor Fall 2013 Jay Taylor (ASU) APM 541 Fall 2013 1 / 53 Outline Outline 1 Introduction 2 Conjugate Distributions 3 Noninformative priors
More informationData Analysis and Uncertainty Part 2: Estimation
Data Analysis and Uncertainty Part 2: Estimation Instructor: Sargur N. University at Buffalo The State University of New York srihari@cedar.buffalo.edu 1 Topics in Estimation 1. Estimation 2. Desirable
More informationBayesian Inference. Chapter 1. Introduction and basic concepts
Bayesian Inference Chapter 1. Introduction and basic concepts M. Concepción Ausín Department of Statistics Universidad Carlos III de Madrid Master in Business Administration and Quantitative Methods Master
More informationPART I INTRODUCTION The meaning of probability Basic definitions for frequentist statistics and Bayesian inference Bayesian inference Combinatorics
Table of Preface page xi PART I INTRODUCTION 1 1 The meaning of probability 3 1.1 Classical definition of probability 3 1.2 Statistical definition of probability 9 1.3 Bayesian understanding of probability
More informationComputer Practical: MetropolisHastingsbased MCMC
Computer Practical: MetropolisHastingsbased MCMC Andrea Arnold and Franz Hamilton North Carolina State University July 30, 2016 A. Arnold / F. Hamilton (NCSU) MHbased MCMC July 30, 2016 1 / 19 Markov
More informationMarkov chain Monte Carlo
Markov chain Monte Carlo Peter Beerli October 10, 2005 [this chapter is highly influenced by chapter 1 in Markov chain Monte Carlo in Practice, eds Gilks W. R. et al. Chapman and Hall/CRC, 1996] 1 Short
More informationSynergistic combination of systems for structural health monitoring and earthquake early warning for structural health prognosis and diagnosis
Synergistic combination of systems for structural health monitoring and earthquake early warning for structural health prognosis and diagnosis Stephen Wu* a, James L. Beck* a a California Institute of
More informationModel Selection for Gaussian Processes
Institute for Adaptive and Neural Computation School of Informatics,, UK December 26 Outline GP basics Model selection: covariance functions and parameterizations Criteria for model selection Marginal
More informationNotes on pseudomarginal methods, variational Bayes and ABC
Notes on pseudomarginal methods, variational Bayes and ABC Christian Andersson Naesseth October 3, 2016 The PseudoMarginal Framework Assume we are interested in sampling from the posterior distribution
More informationTheory and Methods of Statistical Inference
PhD School in Statistics cycle XXIX, 2014 Theory and Methods of Statistical Inference Instructors: B. Liseo, L. Pace, A. Salvan (course coordinator), N. Sartori, A. Tancredi, L. Ventura Syllabus Some prerequisites:
More informationNumerical Analysis for Statisticians
Kenneth Lange Numerical Analysis for Statisticians Springer Contents Preface v 1 Recurrence Relations 1 1.1 Introduction 1 1.2 Binomial CoefRcients 1 1.3 Number of Partitions of a Set 2 1.4 Horner's Method
More informationarxiv:astroph/ v1 14 Sep 2005
For publication in Bayesian Inference and Maximum Entropy Methods, San Jose 25, K. H. Knuth, A. E. Abbas, R. D. Morris, J. P. Castle (eds.), AIP Conference Proceeding A Bayesian Analysis of Extrasolar
More informationMultichannel Deconvolution of Layered Media Using MCMC methods
Multichannel Deconvolution of Layered Media Using MCMC methods Idan Ram Electrical Engineering Department Technion Israel Institute of Technology Supervisors: Prof. Israel Cohen and Prof. Shalom Raz OUTLINE.
More informationSimulated Annealing for Constrained Global Optimization
Monte Carlo Methods for Computation and Optimization Final Presentation Simulated Annealing for Constrained Global Optimization H. Edwin Romeijn & Robert L.Smith (1994) Presented by Ariel Schwartz Objective
More informationBayesian Computations for DSGE Models
Bayesian Computations for DSGE Models Frank Schorfheide University of Pennsylvania, PIER, CEPR, and NBER October 23, 2017 This Lecture is Based on Bayesian Estimation of DSGE Models Edward P. Herbst &
More informationProcess Simulation, Parameter Uncertainty, and Risk
Process Simulation, Parameter Uncertainty, and Risk in QbD and ICH Q8 Design Space John J. Peterson GlaxoSmithKline Pharmaceuticals john.peterson@gsk.com Poor quality Better quality LSL USL Target LSL
More informationStat 451 Lecture Notes Monte Carlo Integration
Stat 451 Lecture Notes 06 12 Monte Carlo Integration Ryan Martin UIC www.math.uic.edu/~rgmartin 1 Based on Chapter 6 in Givens & Hoeting, Chapter 23 in Lange, and Chapters 3 4 in Robert & Casella 2 Updated:
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.utstat.utoronto.ca/~rsalakhu/ Sidney Smith Hall, Room 6002 Lecture 7 Approximate
More informationFinite Element Model Updating Using the Separable Shadow Hybrid Monte Carlo Technique
Finite Element Model Updating Using the Separable Shadow Hybrid Monte Carlo Technique I. Boulkaibet a, L. Mthembu a, T. Marwala a, M. I. Friswell b, S. Adhikari b a The Centre For Intelligent System Modelling
More informationStochastic Backpropagation, Variational Inference, and SemiSupervised Learning
Stochastic Backpropagation, Variational Inference, and SemiSupervised Learning Diederik (Durk) Kingma Danilo J. Rezende (*) Max Welling Shakir Mohamed (**) Stochastic Gradient Variational Inference Bayesian
More informationAdaptive Monte Carlo methods
Adaptive Monte Carlo methods JeanMichel Marin Projet Select, INRIA Futurs, Université ParisSud joint with Randal Douc (École Polytechnique), Arnaud Guillin (Université de Marseille) and Christian Robert
More informationUnsupervised Learning
Unsupervised Learning Bayesian Model Comparison Zoubin Ghahramani zoubin@gatsby.ucl.ac.uk Gatsby Computational Neuroscience Unit, and MSc in Intelligent Systems, Dept Computer Science University College
More informationSequential Monte Carlo Methods
University of Pennsylvania Bradley Visitor Lectures October 23, 2017 Introduction Unfortunately, standard MCMC can be inaccurate, especially in medium and largescale DSGE models: disentangling importance
More informationMonte Carlo integration
Monte Carlo integration Eample of a Monte Carlo sampler in D: imagine a circle radius L/ within a square of LL. If points are randoml generated over the square, what s the probabilit to hit within circle?
More informationBayesian GLMs and MetropolisHastings Algorithm
Bayesian GLMs and MetropolisHastings Algorithm We have seen that with conjugate or semiconjugate prior distributions the Gibbs sampler can be used to sample from the posterior distribution. In situations,
More informationProbabilistic PerformanceBased Optimum Seismic Design of (Bridge) Structures
P E E R U C S D Probabilistic PerformanceBased Optimum Seismic Design of (Bridge) Structures PI: Joel. P. Conte Graduate Student: Yong Li Sponsored by the Pacific Earthquake Engineering Research Center
More informationParticleBased Approximate Inference on Graphical Model
articlebased Approimate Inference on Graphical Model Reference: robabilistic Graphical Model Ch. 2 Koller & Friedman CMU, 0708, Fall 2009 robabilistic Graphical Models Lectures 8,9 Eric ing attern Recognition
More informationStochastic Collocation Methods for Polynomial Chaos: Analysis and Applications
Stochastic Collocation Methods for Polynomial Chaos: Analysis and Applications Dongbin Xiu Department of Mathematics, Purdue University Support: AFOSR FA95581353 (Computational Math) SF CAREER DMS64535
More informationAsymptotically Exact, Embarrassingly Parallel MCMC
Asymptotically Exact, Embarrassingly Parallel MCMC Willie Neiswanger Machine Learning Department Carnegie Mellon University willie@cs.cmu.edu Chong Wang chongw@cs.princeton.edu Eric P. Xing School of Computer
More informationBayesian Statistics: hierarchical models. Sanjib Sharma (University of Sydney)
Bayesian Statistics: hierarchical models Sanjib Sharma (University of Sydney) History of Bayesian statistics Drop a ball on table. Predict its location. Put another ball ask the question is it left or
More informationMetropolis Monte Carlo simulation of the Ising Model
Metropolis Monte Carlo simulation of the Ising Model Krishna Shrinivas (CH10B026) Swaroop Ramaswamy (CH10B068) May 10, 2013 Modelling and Simulation of Particulate Processes (CH5012) Introduction The Ising
More informationHamiltonian Monte Carlo
Hamiltonian Monte Carlo within Stan Daniel Lee Columbia University, Statistics Department bearlee@alum.mit.edu BayesComp mcstan.org Why MCMC? Have data. Have a rich statistical model. No analytic solution.
More informationLearning Bayesian Networks for Biomedical Data
Learning Bayesian Networks for Biomedical Data Faming Liang (Texas A&M University ) Liang, F. and Zhang, J. (2009) Learning Bayesian Networks for Discrete Data. Computational Statistics and Data Analysis,
More informationCALIFORNIA INSTITUTE OF TECHNOLOGY
CALIFORNIA INSTITUTE OF TECHNOLOGY EARTHQUAKE ENGINEERING RESEARCH LABORATORY SEISMIC EARLY WARNING SYSTEMS: PROCEDURE FOR AUTOMATED DECISION MAKING BY VERONICA F. GRASSO, JAMES L. BECK AND GAETANO MANFREDI
More informationA Comparative Study of Imputation Methods for Estimation of Missing Values of Per Capita Expenditure in Central Java
IOP Conference Series: Earth and Environmental Science PAPER OPEN ACCESS A Comparative Study of Imputation Methods for Estimation of Missing Values of Per Capita Expenditure in Central Java To cite this
More informationAnalysis of the Gibbs sampler for a model. related to JamesStein estimators. Jeffrey S. Rosenthal*
Analysis of the Gibbs sampler for a model related to JamesStein estimators by Jeffrey S. Rosenthal* Department of Statistics University of Toronto Toronto, Ontario Canada M5S 1A1 Phone: 416 9784594.
More informationLearning Static Parameters in Stochastic Processes
Learning Static Parameters in Stochastic Processes Bharath Ramsundar December 14, 2012 1 Introduction Consider a Markovian stochastic process X T evolving (perhaps nonlinearly) over time variable T. We
More informationarxiv: v1 [math.st] 4 Dec 2015
MCMC convergence diagnosis using geometry of Bayesian LASSO A. Dermoune, D.Ounaissi, N.Rahmania Abstract arxiv:151.01366v1 [math.st] 4 Dec 015 Using posterior distribution of Bayesian LASSO we construct
More informationA Semiparametric Bayesian Framework for Performance Analysis of Call Centers
Proceedings 59th ISI World Statistics Congress, 2530 August 2013, Hong Kong (Session STS065) p.2345 A Semiparametric Bayesian Framework for Performance Analysis of Call Centers Bangxian Wu and Xiaowei
More informationStatistical Inference and Methods
Department of Mathematics Imperial College London d.stephens@imperial.ac.uk http://stats.ma.ic.ac.uk/ das01/ 31st January 2006 Part VI Session 6: Filtering and Time to Event Data Session 6: Filtering and
More informationMultilevel Statistical Models: 3 rd edition, 2003 Contents
Multilevel Statistical Models: 3 rd edition, 2003 Contents Preface Acknowledgements Notation Two and three level models. A general classification notation and diagram Glossary Chapter 1 An introduction
More informationTHE EFFECT OF THE LATEST SUMATRA EARTHQUAKE TO MALAYSIAN PENINSULAR
JURNAL KEJURUTERAAN AWAM (JOURNAL OF CIVIL ENGINEERING) Vol. 15 No. 2, 2002 THE EFFECT OF THE LATEST SUMATRA EARTHQUAKE TO MALAYSIAN PENINSULAR Assoc. Prof. Dr. Azlan Adnan Hendriyawan Structural Earthquake
More informationAssessment of uncertainty in computer experiments: from Universal Kriging to Bayesian Kriging. Céline Helbert, Delphine Dupuy and Laurent Carraro
Assessment of uncertainty in computer experiments: from Universal Kriging to Bayesian Kriging., Delphine Dupuy and Laurent Carraro Historical context First introduced in the field of geostatistics (Matheron,
More informationA = {(x, u) : 0 u f(x)},
Draw x uniformly from the region {x : f(x) u }. Markov Chain Monte Carlo Lecture 5 Slice sampler: Suppose that one is interested in sampling from a density f(x), x X. Recall that sampling x f(x) is equivalent
More informationIntroduction to Bayesian methods in inverse problems
Introduction to Bayesian methods in inverse problems Ville Kolehmainen 1 1 Department of Applied Physics, University of Eastern Finland, Kuopio, Finland March 4 2013 Manchester, UK. Contents Introduction
More informationELEC633: Graphical Models
ELEC633: Graphical Models Tahira isa Saleem Scribe from 7 October 2008 References: Casella and George Exploring the Gibbs sampler (1992) Chib and Greenberg Understanding the MetropolisHastings algorithm
More informationBayesian Linear Models
Bayesian Linear Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Department of Forestry & Department of Geography, Michigan State University, Lansing Michigan, U.S.A. 2 Biostatistics, School of Public
More informationKarlRudolf Koch Introduction to Bayesian Statistics Second Edition
KarlRudolf Koch Introduction to Bayesian Statistics Second Edition KarlRudolf Koch Introduction to Bayesian Statistics Second, updated and enlarged Edition With 17 Figures Professor Dr.Ing., Dr.Ing.
More informationMeasurement Error and Linear Regression of Astronomical Data. Brandon Kelly Penn State Summer School in Astrostatistics, June 2007
Measurement Error and Linear Regression of Astronomical Data Brandon Kelly Penn State Summer School in Astrostatistics, June 2007 Classical Regression Model Collect n data points, denote i th pair as (η
More informationMarkov Chain Monte Carlo
Markov Chain Monte Carlo Michael Johannes Columbia University Nicholas Polson University of Chicago August 28, 2007 1 Introduction The Bayesian solution to any inference problem is a simple rule: compute
More informationItem Parameter Calibration of LSAT Items Using MCMC Approximation of Bayes Posterior Distributions
R U T C O R R E S E A R C H R E P O R T Item Parameter Calibration of LSAT Items Using MCMC Approximation of Bayes Posterior Distributions Douglas H. Jones a Mikhail Nediak b RRR 72, February, 2! " ##$%#&
More informationMCMC notes by Mark Holder
MCMC notes by Mark Holder Bayesian inference Ultimately, we want to make probability statements about true values of parameters, given our data. For example P(α 0 < α 1 X). According to Bayes theorem:
More informationBayesian Linear Models
Bayesian Linear Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department of Forestry & Department
More informationSampling Algorithms for Probabilistic Graphical models
Sampling Algorithms for Probabilistic Graphical models Vibhav Gogate University of Washington References: Chapter 12 of Probabilistic Graphical models: Principles and Techniques by Daphne Koller and Nir
More information0.1 poisson.bayes: Bayesian Poisson Regression
0.1 poisson.bayes: Bayesian Poisson Regression Use the Poisson regression model if the observations of your dependent variable represents the number of independent events that occur during a fixed period
More informationMolecular Epidemiology Workshop: Bayesian Data Analysis
Molecular Epidemiology Workshop: Bayesian Data Analysis Jay Taylor and Ananias Escalante School of Mathematical and Statistical Sciences Center for Evolutionary Medicine and Informatics Arizona State University
More informationBayesian semiparametric GARCH models
ISSN 1440771X Australia Department of Econometrics and Business Statistics http://www.buseco.monash.edu.au/depts/ebs/pubs/wpapers/ Bayesian semiparametric GARCH models Xibin Zhang and Maxwell L. King
More informationADVANCED FINANCIAL ECONOMETRICS PROF. MASSIMO GUIDOLIN
Massimo Guidolin Massimo.Guidolin@unibocconi.it Dept. of Finance ADVANCED FINANCIAL ECONOMETRICS PROF. MASSIMO GUIDOLIN a.a. 14/15 p. 1 LECTURE 3: REVIEW OF BASIC ESTIMATION METHODS: GMM AND OTHER EXTREMUM
More informationMachine Learning. Lecture 4: Regularization and Bayesian Statistics. Feng Li. https://funglee.github.io
Machine Learning Lecture 4: Regularization and Bayesian Statistics Feng Li fli@sdu.edu.cn https://funglee.github.io School of Computer Science and Technology Shandong University Fall 207 Overfitting Problem
More informationQuantile POD for HitMiss Data
Quantile POD for HitMiss Data YewMeng Koh a and William Q. Meeker a a Center for Nondestructive Evaluation, Department of Statistics, Iowa State niversity, Ames, Iowa 50010 Abstract. Probability of detection
More informationLecture 2: From Linear Regression to Kalman Filter and Beyond
Lecture 2: From Linear Regression to Kalman Filter and Beyond Department of Biomedical Engineering and Computational Science Aalto University January 26, 2012 Contents 1 Batch and Recursive Estimation
More informationSpatial discrete hazards using Hierarchical Bayesian Modeling
Spatial discrete hazards using Hierarchical Bayesian Modeling Mathias Graf ETH Zurich, Institute for Structural Engineering, Group Risk & Safety 1 Papers Maes, M.A., Dann M., Sarkar S., and Midtgaard,
More informationCovariance Matrix Simplification For Efficient Uncertainty Management
PASEO MaxEnt 2007 Covariance Matrix Simplification For Efficient Uncertainty Management André Jalobeanu, Jorge A. Gutiérrez PASEO Research Group LSIIT (CNRS/ Univ. Strasbourg)  Illkirch, France *part
More informationThe Origin of Deep Learning. Lili Mou Jan, 2015
The Origin of Deep Learning Lili Mou Jan, 2015 Acknowledgment Most of the materials come from G. E. Hinton s online course. Outline Introduction Preliminary Boltzmann Machines and RBMs Deep Belief Nets
More informationAn introduction to Sequential Monte Carlo
An introduction to Sequential Monte Carlo Thang Bui Jes Frellsen Department of Engineering University of Cambridge Research and Communication Club 6 February 2014 1 Sequential Monte Carlo (SMC) methods
More informationNotes on Noise Contrastive Estimation (NCE)
Notes on Noise Contrastive Estimation NCE) David Meyer dmm@{45.net,uoregon.edu,...} March 0, 207 Introduction In this note we follow the notation used in [2]. Suppose X x, x 2,, x Td ) is a sample of
More informationSTA 294: Stochastic Processes & Bayesian Nonparametrics
MARKOV CHAINS AND CONVERGENCE CONCEPTS Markov chains are among the simplest stochastic processes, just one step beyond iid sequences of random variables. Traditionally they ve been used in modelling a
More informationESTIMATION of a DSGE MODEL
ESTIMATION of a DSGE MODEL Paris, October 17 2005 STÉPHANE ADJEMIAN stephane.adjemian@ens.fr UNIVERSITÉ DU MAINE & CEPREMAP Slides.tex ESTIMATION of a DSGE MODEL STÉPHANE ADJEMIAN 16/10/2005 21:37 p. 1/3
More informationGeoMarine Letters Volume 36, 2016, electronic supplementary material
1 GeoMarine Letters Volume 36, 016, electronic supplementary material Submarine landslides offshore Vancouver Island along the northern Cascadia margin, British Columbia: why preconditioning is likely
More informationThe Generalized Likelihood Uncertainty Estimation methodology
CHAPTER 4 The Generalized Likelihood Uncertainty Estimation methodology Calibration and uncertainty estimation based upon a statistical framework is aimed at finding an optimal set of models, parameters
More informationCS 630 Basic Probability and Information Theory. Tim Campbell
CS 630 Basic Probability and Information Theory Tim Campbell 21 January 2003 Probability Theory Probability Theory is the study of how best to predict outcomes of events. An experiment (or trial or event)
More informationStat 451 Lecture Notes Numerical Integration
Stat 451 Lecture Notes 03 12 Numerical Integration Ryan Martin UIC www.math.uic.edu/~rgmartin 1 Based on Chapter 5 in Givens & Hoeting, and Chapters 4 & 18 of Lange 2 Updated: February 11, 2016 1 / 29
More informationBayesian Learning. HT2015: SC4 Statistical Data Mining and Machine Learning. Maximum Likelihood Principle. The Bayesian Learning Framework
HT5: SC4 Statistical Data Mining and Machine Learning Dino Sejdinovic Department of Statistics Oxford http://www.stats.ox.ac.uk/~sejdinov/sdmml.html Maximum Likelihood Principle A generative model for
More information