Model Choice. Hoff Chapter 9. Dec 8, 2010
|
|
- Rafe Simon
- 6 years ago
- Views:
Transcription
1 Model Choice Hoff Chapter 9 Dec 8, 2010
2 Topics Variable Selection / Model Choice Stepwise Methods Model Selection Criteria Model Averaging
3 Variable Selection Reasons for reducing the number of variables in the model: Philosophical Avoid the use of redundant variables (problems with interpretations) KISS Occam s Razor Practical Inclusion of un-necessary terms yields less precise estimates, particularly if explanatory variables are highly correlated with each other
4 Variable Selection Procedures Stepwise Regression: Forward, Stepwise, Backward add/delete variables until all t-statistics are significant (easy, but not recommended) Use a Model Selection Criterion to pick the best model R2 (picks largest model) Adjusted R2 Mallow s Cp Cp = (SSE/ˆσ Full 2 ) + 2p m n AIC (Akaike Information Criterion) proportional to Cp for linear models BIC(m) (Bayes Information Criterion) ˆσ 2 m + log(n)p m Trade off model complexity (number of coefficients p m ) with goodness of fit ( ˆσ 2 m)
5 Model Selection Selection of a single model has the following problems When the criteria suggest that several models are equally good, what should we report? Still pick only one model? What do we report for our uncertainty after selecting a model? Typical analysis ignores model uncertainty!
6 Bayesian Model Choice Models for the variable selection problem are based on a subset of the X 1,... X p variables Encode models with a vector γ = (γ 1,... γ p ) where γ j {0, 1} is an indicator for whether variable X j should be included in the model M γ. γ j = 0 β j = 0 Each value of γ represents one of the 2 p models. Under model M γ : Y β, σ 2, γ N(X γ β γ, σ 2 I ) Where X γ is design matrix using the columns in X where γ j = 1 and β γ is the subset of β that are non-zero.
7 Bayesian Model Averaging Rather than use a single model, BMA uses all (or potentially a lot) models, but weights model predictions by their posterior probabilities (measure of how much each model is supported by the data) Posterior model probabilities p(m j Y) = p(y M j)p(m j ) j p(y M j)p(m j ) Marginal likelihod of a model is proportional to p(y M γ ) = p(y β γ, σ 2 )p(β γ γ, σ 2 )p(σ 2 γ)dβ dσ 2 Probability β j 0: M j :β j 0 p(m j Y) Predictions ˆ Y Y = j p(m j Y)ŶM j
8 Prior Distributions Bayesian Model choice requires proper prior distributions on regression coefficients Vague but proper priors may lead to paradoxes! Conjugate Normal-Gammas lead to closed form expressions for marginal likelihoods, Zellner s g-prior is the most popular.
9 Zellner s g-prior Centered model: Y = 1 n α + X c β + ɛ where X c is the centered design matrix where all variables have had their mean subtracted p(α) 1 p(σ 2 ) 1/σ 2 β γ σ 2 γ N(0, gσ 2 (X c X c ) 1 ) take g = n which leads to marginal likelihood of M γ that is proportional to p(y M γ ) = C(1 + g) n p 1 2 (1 + g(1 Rγ 2 (n 1) )) 2 where R 2 is the usual R 2 for model M γ. Trade-off of model complexity versus goodness of fit Lastly, assign uniform distribution to space of models
10 USair Data library(bas) poll.bma = bas.lm(log(so2) ~ temp + log(firms) + log(popn) + wind + precip+ rain, data=pollution, prior="g-prior", alpha=41, n.models=2^7, update=50, initprobs="uniform") par(mfrow=c(2,2)) plot(poll.bma, ask=f)
11 Plots Residuals vs Fitted Model Probabilities Residuals Cumulative Probability Predictions under BMA Model Search Order Model Complexity Inclusion Probabilities log(marginal) Marginal Inclusion Probability Model Dimension Intercept temp log(firms) log(popn) wind precip rain
12 Model Space image(poll.bma) Intercept temp log(firms) log(popn) wind precip rain Log Posterior Odds Model Rank
13 Coefficients beta = coef(poll.bma) par(mfrow=c(2,3)); plot(beta, subset=2:7,ask=f) temp log(firms) log(popn) wind precip rain
14 Mortality & Pollution Data from Statistical Sleuth cities response Mortality measures of HC, NOX, SO2 Is pollution associated with mortality after adjusting for other socio-economic and meteorological factors? 15 predictor variables implies 2 15 = 32, 768 possible models
15 5 Model Dimension DENSITY Model Complexity Inclusion Probabilities logso loghc lognox Model Search Order POOR Predictions under BMA WHITECOL NONWHITE EDUC 5000 SOUND HOUSE Residuals Cumulative Probability Residuals vs Fitted OVER JANTEMP JULYTEMP 25 4 PRECIP HUMIDITY Intercept 10 log(marginal) 5 Marginal Inclusion Probability Posterior Distributions 60 Model Probabilities 30000
16 Model Space Intercept PRECIP HUMIDITY JANTEMP JULYTEMP OVER65 HOUSE EDUC SOUND DENSITY NONWHITE WHITECOL POOR loghc lognox logso2 Log Posterior Odds Model Rank
17 Coefficients Intercept PRECIP HUMIDITY JANTEMP JULYTEMP OVER65 HOUSE EDUC
18 Coefficients SOUND DENSITY NONWHITE WHITECOL POOR loghc lognox logso
19 Posterior Probabilities What is the probability that there is no pollution effect? Sum posterior model probabilities over all models that include no pollution variables which.mat = list2matrix.which(mort.bma,1:(2^15)) poll.in = (which.mat[, 14:16] %*% rep(1, 3)) == 0 sum(poll.in * mort.bma$postprob) Posterior probability is Odds that there is an effect (1.011)/(.011) = 88.6 Prior Odds 7.9 = (1.2 3 )/.2 3 Bayes Factor for a pollution effect 88.6/7.9 = 11.2 Bayes Factor for an NOX 0.918/( ) = 11.2 Bayes Factors are not monotonic!
20 Problems if p > enumeration is difficult Gibbs sampler on γ poor convergence/mixing with high correlations Metropolis Hastings algorithms more flexibility Stochastic Search (no guarantee samples represent posterior) in BMA all variables are included, but coefficients are shrunk to 0; alternative is to use Shrinkage methods Choice of prior distributions on β and on γ Model averaging versus Model Selection what are objectives?
Model Choice. Hoff Chapter 9, Clyde & George Model Uncertainty StatSci, Hoeting et al BMA StatSci. October 27, 2015
Model Choice Hoff Chapter 9, Clyde & George Model Uncertainty StatSci, Hoeting et al BMA StatSci October 27, 2015 Topics Variable Selection / Model Choice Stepwise Methods Model Selection Criteria Model
More informationMixtures of Prior Distributions
Mixtures of Prior Distributions Hoff Chapter 9, Liang et al 2007, Hoeting et al (1999), Clyde & George (2004) November 10, 2016 Bartlett s Paradox The Bayes factor for comparing M γ to the null model:
More informationMixtures of Prior Distributions
Mixtures of Prior Distributions Hoff Chapter 9, Liang et al 2007, Hoeting et al (1999), Clyde & George (2004) November 9, 2017 Bartlett s Paradox The Bayes factor for comparing M γ to the null model: BF
More informationBayesian Model Averaging
Bayesian Model Averaging Hoff Chapter 9, Hoeting et al 1999, Clyde & George 2004, Liang et al 2008 October 24, 2017 Bayesian Model Choice Models for the variable selection problem are based on a subset
More informationMODEL AVERAGING by Merlise Clyde 1
Chapter 13 MODEL AVERAGING by Merlise Clyde 1 13.1 INTRODUCTION In Chapter 12, we considered inference in a normal linear regression model with q predictors. In many instances, the set of predictor variables
More informationNotes for ISyE 6413 Design and Analysis of Experiments
Notes for ISyE 6413 Design and Analysis of Experiments Instructor : C. F. Jeff Wu School of Industrial and Systems Engineering Georgia Institute of Technology Text book : Experiments : Planning, Analysis,
More informationRegression, Ridge Regression, Lasso
Regression, Ridge Regression, Lasso Fabio G. Cozman - fgcozman@usp.br October 2, 2018 A general definition Regression studies the relationship between a response variable Y and covariates X 1,..., X n.
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 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 information10. Alternative case influence statistics
10. Alternative case influence statistics a. Alternative to D i : dffits i (and others) b. Alternative to studres i : externally-studentized residual c. Suggestion: use whatever is convenient with the
More informationST440/540: Applied Bayesian Statistics. (9) Model selection and goodness-of-fit checks
(9) Model selection and goodness-of-fit checks Objectives In this module we will study methods for model comparisons and checking for model adequacy For model comparisons there are a finite number of candidate
More informationThe linear model is the most fundamental of all serious statistical models encompassing:
Linear Regression Models: A Bayesian perspective Ingredients of a linear model include an n 1 response vector y = (y 1,..., y n ) T and an n p design matrix (e.g. including regressors) X = [x 1,..., x
More informationSparse Linear Models (10/7/13)
STA56: Probabilistic machine learning Sparse Linear Models (0/7/) Lecturer: Barbara Engelhardt Scribes: Jiaji Huang, Xin Jiang, Albert Oh Sparsity Sparsity has been a hot topic in statistics and machine
More informationMarcel Dettling. Applied Statistical Regression AS 2012 Week 05. ETH Zürich, October 22, Institute for Data Analysis and Process Design
Marcel Dettling Institute for Data Analysis and Process Design Zurich University of Applied Sciences marcel.dettling@zhaw.ch http://stat.ethz.ch/~dettling ETH Zürich, October 22, 2012 1 What is Regression?
More informationBayesian methods in economics and finance
1/26 Bayesian methods in economics and finance Linear regression: Bayesian model selection and sparsity priors Linear Regression 2/26 Linear regression Model for relationship between (several) independent
More informationSTAT 425: Introduction to Bayesian Analysis
STAT 425: Introduction to Bayesian Analysis Marina Vannucci Rice University, USA Fall 2017 Marina Vannucci (Rice University, USA) Bayesian Analysis (Part 3) Fall 2017 1 / 40 Part 3: Hierarchical and Linear
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 informationIntroduction to Statistical modeling: handout for Math 489/583
Introduction to Statistical modeling: handout for Math 489/583 Statistical modeling occurs when we are trying to model some data using statistical tools. From the start, we recognize that no model is perfect
More informationHow the mean changes depends on the other variable. Plots can show what s happening...
Chapter 8 (continued) Section 8.2: Interaction models An interaction model includes one or several cross-product terms. Example: two predictors Y i = β 0 + β 1 x i1 + β 2 x i2 + β 12 x i1 x i2 + ɛ i. How
More informationBayesian Linear Regression
Bayesian Linear Regression Sudipto Banerjee 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. September 15, 2010 1 Linear regression models: a Bayesian perspective
More informationModel Selection. Frank Wood. December 10, 2009
Model Selection Frank Wood December 10, 2009 Standard Linear Regression Recipe Identify the explanatory variables Decide the functional forms in which the explanatory variables can enter the model Decide
More informationBayesian Linear Models
Bayesian Linear Models Sudipto Banerjee September 03 05, 2017 Department of Biostatistics, Fielding School of Public Health, University of California, Los Angeles Linear Regression Linear regression is,
More informationSTAT 100C: Linear models
STAT 100C: Linear models Arash A. Amini June 9, 2018 1 / 21 Model selection Choosing the best model among a collection of models {M 1, M 2..., M N }. What is a good model? 1. fits the data well (model
More informationBayesian Machine Learning
Bayesian Machine Learning Andrew Gordon Wilson ORIE 6741 Lecture 3 Stochastic Gradients, Bayesian Inference, and Occam s Razor https://people.orie.cornell.edu/andrew/orie6741 Cornell University August
More informationAMS-207: Bayesian Statistics
Linear Regression How does a quantity y, vary as a function of another quantity, or vector of quantities x? We are interested in p(y θ, x) under a model in which n observations (x i, y i ) are exchangeable.
More informationMachine Learning for OR & FE
Machine Learning for OR & FE Regression II: Regularization and Shrinkage Methods Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationVariable Selection in Predictive Regressions
Variable Selection in Predictive Regressions Alessandro Stringhi Advanced Financial Econometrics III Winter/Spring 2018 Overview This chapter considers linear models for explaining a scalar variable when
More informationConsistent high-dimensional Bayesian variable selection via penalized credible regions
Consistent high-dimensional Bayesian variable selection via penalized credible regions Howard Bondell bondell@stat.ncsu.edu Joint work with Brian Reich Howard Bondell p. 1 Outline High-Dimensional Variable
More informationRegression I: Mean Squared Error and Measuring Quality of Fit
Regression I: Mean Squared Error and Measuring Quality of Fit -Applied Multivariate Analysis- Lecturer: Darren Homrighausen, PhD 1 The Setup Suppose there is a scientific problem we are interested in solving
More informationDay 4: Shrinkage Estimators
Day 4: Shrinkage Estimators Kenneth Benoit Data Mining and Statistical Learning March 9, 2015 n versus p (aka k) Classical regression framework: n > p. Without this inequality, the OLS coefficients have
More informationContents. Part I: Fundamentals of Bayesian Inference 1
Contents Preface xiii Part I: Fundamentals of Bayesian Inference 1 1 Probability and inference 3 1.1 The three steps of Bayesian data analysis 3 1.2 General notation for statistical inference 4 1.3 Bayesian
More informationModule 11: Linear Regression. Rebecca C. Steorts
Module 11: Linear Regression Rebecca C. Steorts Announcements Today is the last class Homework 7 has been extended to Thursday, April 20, 11 PM. There will be no lab tomorrow. There will be office hours
More informationMS-C1620 Statistical inference
MS-C1620 Statistical inference 10 Linear regression III Joni Virta Department of Mathematics and Systems Analysis School of Science Aalto University Academic year 2018 2019 Period III - IV 1 / 32 Contents
More informationA Frequentist Assessment of Bayesian Inclusion Probabilities
A Frequentist Assessment of Bayesian Inclusion Probabilities Department of Statistical Sciences and Operations Research October 13, 2008 Outline 1 Quantitative Traits Genetic Map The model 2 Bayesian Model
More informationHypothesis Testing. Econ 690. Purdue University. Justin L. Tobias (Purdue) Testing 1 / 33
Hypothesis Testing Econ 690 Purdue University Justin L. Tobias (Purdue) Testing 1 / 33 Outline 1 Basic Testing Framework 2 Testing with HPD intervals 3 Example 4 Savage Dickey Density Ratio 5 Bartlett
More informationFinal Review. Yang Feng. Yang Feng (Columbia University) Final Review 1 / 58
Final Review Yang Feng http://www.stat.columbia.edu/~yangfeng Yang Feng (Columbia University) Final Review 1 / 58 Outline 1 Multiple Linear Regression (Estimation, Inference) 2 Special Topics for Multiple
More informationPenalized Loss functions for Bayesian Model Choice
Penalized Loss functions for Bayesian Model Choice Martyn International Agency for Research on Cancer Lyon, France 13 November 2009 The pure approach For a Bayesian purist, all uncertainty is represented
More informationLinear Regression In God we trust, all others bring data. William Edwards Deming
Linear Regression ddebarr@uw.edu 2017-01-19 In God we trust, all others bring data. William Edwards Deming Course Outline 1. Introduction to Statistical Learning 2. Linear Regression 3. Classification
More informationLinear Models (continued)
Linear Models (continued) Model Selection Introduction Most of our previous discussion has focused on the case where we have a data set and only one fitted model. Up until this point, we have discussed,
More informationStandard Errors & Confidence Intervals. N(0, I( β) 1 ), I( β) = [ 2 l(β, φ; y) β i β β= β j
Standard Errors & Confidence Intervals β β asy N(0, I( β) 1 ), where I( β) = [ 2 l(β, φ; y) ] β i β β= β j We can obtain asymptotic 100(1 α)% confidence intervals for β j using: β j ± Z 1 α/2 se( β j )
More informationLinear Models A linear model is defined by the expression
Linear Models A linear model is defined by the expression x = F β + ɛ. where x = (x 1, x 2,..., x n ) is vector of size n usually known as the response vector. β = (β 1, β 2,..., β p ) is the transpose
More informationDefault Priors and Effcient Posterior Computation in Bayesian
Default Priors and Effcient Posterior Computation in Bayesian Factor Analysis January 16, 2010 Presented by Eric Wang, Duke University Background and Motivation A Brief Review of Parameter Expansion Literature
More informationLinear Regression Models. Based on Chapter 3 of Hastie, Tibshirani and Friedman
Linear Regression Models Based on Chapter 3 of Hastie, ibshirani and Friedman Linear Regression Models Here the X s might be: p f ( X = " + " 0 j= 1 X j Raw predictor variables (continuous or coded-categorical
More informationGibbs Sampling in Linear Models #1
Gibbs Sampling in Linear Models #1 Econ 690 Purdue University Justin L Tobias Gibbs Sampling #1 Outline 1 Conditional Posterior Distributions for Regression Parameters in the Linear Model [Lindley and
More informationPower-Expected-Posterior Priors for Variable Selection in Gaussian Linear Models
Power-Expected-Posterior Priors for Variable Selection in Gaussian Linear Models Dimitris Fouskakis, Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical
More informationHigh-dimensional regression modeling
High-dimensional regression modeling David Causeur Department of Statistics and Computer Science Agrocampus Ouest IRMAR CNRS UMR 6625 http://www.agrocampus-ouest.fr/math/causeur/ Course objectives Making
More informationStatistics 262: Intermediate Biostatistics Model selection
Statistics 262: Intermediate Biostatistics Model selection Jonathan Taylor & Kristin Cobb Statistics 262: Intermediate Biostatistics p.1/?? Today s class Model selection. Strategies for model selection.
More informationPart 7: Hierarchical Modeling
Part 7: Hierarchical Modeling!1 Nested data It is common for data to be nested: i.e., observations on subjects are organized by a hierarchy Such data are often called hierarchical or multilevel For example,
More informationStat/F&W Ecol/Hort 572 Review Points Ané, Spring 2010
1 Linear models Y = Xβ + ɛ with ɛ N (0, σ 2 e) or Y N (Xβ, σ 2 e) where the model matrix X contains the information on predictors and β includes all coefficients (intercept, slope(s) etc.). 1. Number of
More informationModel comparison and selection
BS2 Statistical Inference, Lectures 9 and 10, Hilary Term 2008 March 2, 2008 Hypothesis testing Consider two alternative models M 1 = {f (x; θ), θ Θ 1 } and M 2 = {f (x; θ), θ Θ 2 } for a sample (X = x)
More informationModel Choice. Brian D. Ripley. Professor of Applied Statistics University of Oxford.
Model Choice Brian D. Ripley Professor of Applied Statistics University of Oxford ripley@stats.ox.ac.uk http://stats.ox.ac.uk/ ripley Sub-titles Towards general principles for model selection. Give up
More informationSubject CS1 Actuarial Statistics 1 Core Principles
Institute of Actuaries of India Subject CS1 Actuarial Statistics 1 Core Principles For 2019 Examinations Aim The aim of the Actuarial Statistics 1 subject is to provide a grounding in mathematical and
More information5. Erroneous Selection of Exogenous Variables (Violation of Assumption #A1)
5. Erroneous Selection of Exogenous Variables (Violation of Assumption #A1) Assumption #A1: Our regression model does not lack of any further relevant exogenous variables beyond x 1i, x 2i,..., x Ki and
More informationStatistical Models. ref: chapter 1 of Bates, D and D. Watts (1988) Nonlinear Regression Analysis and its Applications, Wiley. Dave Campbell 2009
Statistical Models ref: chapter 1 of Bates, D and D. Watts (1988) Nonlinear Regression Analysis and its Applications, Wiley Dave Campbell 2009 Today linear regression in terms of the response surface geometry
More informationvariability of the model, represented by σ 2 and not accounted for by Xβ
Posterior Predictive Distribution Suppose we have observed a new set of explanatory variables X and we want to predict the outcomes ỹ using the regression model. Components of uncertainty in p(ỹ y) variability
More informationVector Autoregressive Model. Vector Autoregressions II. Estimation of Vector Autoregressions II. Estimation of Vector Autoregressions I.
Vector Autoregressive Model Vector Autoregressions II Empirical Macroeconomics - Lect 2 Dr. Ana Beatriz Galvao Queen Mary University of London January 2012 A VAR(p) model of the m 1 vector of time series
More informationRonald Christensen. University of New Mexico. Albuquerque, New Mexico. Wesley Johnson. University of California, Irvine. Irvine, California
Texts in Statistical Science Bayesian Ideas and Data Analysis An Introduction for Scientists and Statisticians Ronald Christensen University of New Mexico Albuquerque, New Mexico Wesley Johnson University
More informationMixtures of g-priors for Bayesian Variable Selection
Mixtures of g-priors for Bayesian Variable Selection January 8, 007 Abstract Zellner s g-prior remains a popular conventional prior for use in Bayesian variable selection, despite several undesirable consistency
More informationVariable Selection and Model Building
LINEAR REGRESSION ANALYSIS MODULE XIII Lecture - 39 Variable Selection and Model Building Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur 5. Akaike s information
More informationTechnical Appendix Detailing Estimation Procedures used in A. Flexible Approach to Modeling Ultimate Recoveries on Defaulted
Technical Appendix Detailing Estimation Procedures used in A Flexible Approach to Modeling Ultimate Recoveries on Defaulted Loans and Bonds (Web Supplement) Section 1 of this Appendix provides a brief
More informationLinear Regression Models
Linear Regression Models Model Description and Model Parameters Modelling is a central theme in these notes. The idea is to develop and continuously improve a library of predictive models for hazards,
More informationPart 6: Multivariate Normal and Linear Models
Part 6: Multivariate Normal and Linear Models 1 Multiple measurements Up until now all of our statistical models have been univariate models models for a single measurement on each member of a sample of
More informationTransformations The bias-variance tradeoff Model selection criteria Remarks. Model selection I. Patrick Breheny. February 17
Model selection I February 17 Remedial measures Suppose one of your diagnostic plots indicates a problem with the model s fit or assumptions; what options are available to you? Generally speaking, you
More informationBayesian inference. Rasmus Waagepetersen Department of Mathematics Aalborg University Denmark. April 10, 2017
Bayesian inference Rasmus Waagepetersen Department of Mathematics Aalborg University Denmark April 10, 2017 1 / 22 Outline for today A genetic example Bayes theorem Examples Priors Posterior summaries
More informationNovember 2002 STA Random Effects Selection in Linear Mixed Models
November 2002 STA216 1 Random Effects Selection in Linear Mixed Models November 2002 STA216 2 Introduction It is common practice in many applications to collect multiple measurements on a subject. Linear
More informationBayesian Methods in Multilevel Regression
Bayesian Methods in Multilevel Regression Joop Hox MuLOG, 15 september 2000 mcmc What is Statistics?! Statistics is about uncertainty To err is human, to forgive divine, but to include errors in your design
More informationMachine Learning for Biomedical Engineering. Enrico Grisan
Machine Learning for Biomedical Engineering Enrico Grisan enrico.grisan@dei.unipd.it Curse of dimensionality Why are more features bad? Redundant features (useless or confounding) Hard to interpret and
More informationISyE 691 Data mining and analytics
ISyE 691 Data mining and analytics Regression Instructor: Prof. Kaibo Liu Department of Industrial and Systems Engineering UW-Madison Email: kliu8@wisc.edu Office: Room 3017 (Mechanical Engineering Building)
More informationLecture 7: Modeling Krak(en)
Lecture 7: Modeling Krak(en) Variable selection Last In both time cases, we saw we two are left selection with selection problems problem -- How -- do How we pick do we either pick either subset the of
More informationMLR Model Selection. Author: Nicholas G Reich, Jeff Goldsmith. This material is part of the statsteachr project
MLR Model Selection Author: Nicholas G Reich, Jeff Goldsmith This material is part of the statsteachr project Made available under the Creative Commons Attribution-ShareAlike 3.0 Unported License: http://creativecommons.org/licenses/by-sa/3.0/deed.en
More informationLecture 4: Dynamic models
linear s Lecture 4: s Hedibert Freitas Lopes The University of Chicago Booth School of Business 5807 South Woodlawn Avenue, Chicago, IL 60637 http://faculty.chicagobooth.edu/hedibert.lopes hlopes@chicagobooth.edu
More informationLinear Model Selection and Regularization
Linear Model Selection and Regularization Recall the linear model Y = β 0 + β 1 X 1 + + β p X p + ɛ. In the lectures that follow, we consider some approaches for extending the linear model framework. In
More informationRobust Bayesian Regression
Readings: Hoff Chapter 9, West JRSSB 1984, Fúquene, Pérez & Pericchi 2015 Duke University November 17, 2016 Body Fat Data: Intervals w/ All Data Response % Body Fat and Predictor Waist Circumference 95%
More informationDynamic Generalized Linear Models
Dynamic Generalized Linear Models Jesse Windle Oct. 24, 2012 Contents 1 Introduction 1 2 Binary Data (Static Case) 2 3 Data Augmentation (de-marginalization) by 4 examples 3 3.1 Example 1: CDF method.............................
More informationECLT 5810 Linear Regression and Logistic Regression for Classification. Prof. Wai Lam
ECLT 5810 Linear Regression and Logistic Regression for Classification Prof. Wai Lam Linear Regression Models Least Squares Input vectors is an attribute / feature / predictor (independent variable) The
More informationFEEG6017 lecture: Akaike's information criterion; model reduction. Brendan Neville
FEEG6017 lecture: Akaike's information criterion; model reduction Brendan Neville bjn1c13@ecs.soton.ac.uk Occam's razor William of Occam, 1288-1348. All else being equal, the simplest explanation is the
More informationSupplementary materials for Scalable Bayesian model averaging through local information propagation
Supplementary materials for Scalable Bayesian model averaging through local information propagation August 25, 2014 S1. Proofs Proof of Theorem 1. The result follows immediately from the distributions
More informationRegression modeling for categorical data. Part II : Model selection and prediction
Regression modeling for categorical data Part II : Model selection and prediction David Causeur Agrocampus Ouest IRMAR CNRS UMR 6625 http://math.agrocampus-ouest.fr/infogluedeliverlive/membres/david.causeur
More informationModelling Operational Risk Using Bayesian Inference
Pavel V. Shevchenko Modelling Operational Risk Using Bayesian Inference 4y Springer 1 Operational Risk and Basel II 1 1.1 Introduction to Operational Risk 1 1.2 Defining Operational Risk 4 1.3 Basel II
More informationData-Driven Bayesian Model Selection: Parameter Space Dimension Reduction using Automatic Relevance Determination Priors
Data-Driven : Parameter Space Dimension Reduction using Priors Mohammad Khalil mkhalil@sandia.gov, Livermore, CA Workshop on Uncertainty Quantification and Data-Driven Modeling Austin, Texas March 23-24,
More informationBayesian Regression (1/31/13)
STA613/CBB540: Statistical methods in computational biology Bayesian Regression (1/31/13) Lecturer: Barbara Engelhardt Scribe: Amanda Lea 1 Bayesian Paradigm Bayesian methods ask: given that I have observed
More informationComputer Vision Group Prof. Daniel Cremers. 10a. Markov Chain Monte Carlo
Group Prof. Daniel Cremers 10a. Markov Chain Monte Carlo Markov Chain Monte Carlo In high-dimensional spaces, rejection sampling and importance sampling are very inefficient An alternative is Markov Chain
More informationExample: Ground Motion Attenuation
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
More informationModel Selection in GLMs. (should be able to implement frequentist GLM analyses!) Today: standard frequentist methods for model selection
Model Selection in GLMs Last class: estimability/identifiability, analysis of deviance, standard errors & confidence intervals (should be able to implement frequentist GLM analyses!) Today: standard frequentist
More informationMarkov Chain Monte Carlo methods
Markov Chain Monte Carlo methods By Oleg Makhnin 1 Introduction a b c M = d e f g h i 0 f(x)dx 1.1 Motivation 1.1.1 Just here Supresses numbering 1.1.2 After this 1.2 Literature 2 Method 2.1 New math As
More informationMCMC algorithms for fitting Bayesian models
MCMC algorithms for fitting Bayesian models p. 1/1 MCMC algorithms for fitting Bayesian models Sudipto Banerjee sudiptob@biostat.umn.edu University of Minnesota MCMC algorithms for fitting Bayesian models
More informationBusiness Statistics. Tommaso Proietti. Model Evaluation and Selection. DEF - Università di Roma 'Tor Vergata'
Business Statistics Tommaso Proietti DEF - Università di Roma 'Tor Vergata' Model Evaluation and Selection Predictive Ability of a Model: Denition and Estimation We aim at achieving a balance between parsimony
More informationDEM Working Paper Series. Credit risk predictions with Bayesian model averaging
ISSN: 2281-1346 Department of Economics and Management DEM Working Paper Series Credit risk predictions with Bayesian model averaging Silvia Figini (Università di Pavia) Paolo Giudici (Università di Pavia)
More information7. Estimation and hypothesis testing. Objective. Recommended reading
7. Estimation and hypothesis testing Objective In this chapter, we show how the election of estimators can be represented as a decision problem. Secondly, we consider the problem of hypothesis testing
More informationCS 6140: Machine Learning Spring What We Learned Last Week. Survey 2/26/16. VS. Model
Logis@cs CS 6140: Machine Learning Spring 2016 Instructor: Lu Wang College of Computer and Informa@on Science Northeastern University Webpage: www.ccs.neu.edu/home/luwang Email: luwang@ccs.neu.edu Assignment
More informationBayesian Methods for Machine Learning
Bayesian Methods for Machine Learning CS 584: Big Data Analytics Material adapted from Radford Neal s tutorial (http://ftp.cs.utoronto.ca/pub/radford/bayes-tut.pdf), Zoubin Ghahramni (http://hunch.net/~coms-4771/zoubin_ghahramani_bayesian_learning.pdf),
More informationMachine Learning Linear Regression. Prof. Matteo Matteucci
Machine Learning Linear Regression Prof. Matteo Matteucci Outline 2 o Simple Linear Regression Model Least Squares Fit Measures of Fit Inference in Regression o Multi Variate Regession Model Least Squares
More informationCS 6140: Machine Learning Spring 2016
CS 6140: Machine Learning Spring 2016 Instructor: Lu Wang College of Computer and Informa?on Science Northeastern University Webpage: www.ccs.neu.edu/home/luwang Email: luwang@ccs.neu.edu Logis?cs Assignment
More informationLinear Regression With Special Variables
Linear Regression With Special Variables Junhui Qian December 21, 2014 Outline Standardized Scores Quadratic Terms Interaction Terms Binary Explanatory Variables Binary Choice Models Standardized Scores:
More information1 Introduction. 2 AIC versus SBIC. Erik Swanson Cori Saviano Li Zha Final Project
Erik Swanson Cori Saviano Li Zha Final Project 1 Introduction In analyzing time series data, we are posed with the question of how past events influences the current situation. In order to determine this,
More informationBIO5312 Biostatistics Lecture 13: Maximum Likelihood Estimation
BIO5312 Biostatistics Lecture 13: Maximum Likelihood Estimation Yujin Chung November 29th, 2016 Fall 2016 Yujin Chung Lec13: MLE Fall 2016 1/24 Previous Parametric tests Mean comparisons (normality assumption)
More informationST 740: Model Selection
ST 740: Model Selection Alyson Wilson Department of Statistics North Carolina State University November 25, 2013 A. Wilson (NCSU Statistics) Model Selection November 25, 2013 1 / 29 Formal Bayesian Model
More informationBiostatistics Advanced Methods in Biostatistics IV
Biostatistics 140.754 Advanced Methods in Biostatistics IV Jeffrey Leek Assistant Professor Department of Biostatistics jleek@jhsph.edu Lecture 12 1 / 36 Tip + Paper Tip: As a statistician the results
More informationBayesian Inference: Probit and Linear Probability Models
Utah State University DigitalCommons@USU All Graduate Plan B and other Reports Graduate Studies 5-1-2014 Bayesian Inference: Probit and Linear Probability Models Nate Rex Reasch Utah State University Follow
More informationStat 5101 Lecture Notes
Stat 5101 Lecture Notes Charles J. Geyer Copyright 1998, 1999, 2000, 2001 by Charles J. Geyer May 7, 2001 ii Stat 5101 (Geyer) Course Notes Contents 1 Random Variables and Change of Variables 1 1.1 Random
More information