Session 2B: Some basic simulation methods
|
|
- Carol Lane
- 5 years ago
- Views:
Transcription
1 Session 2B: Some basic simulation methods John Geweke Bayesian Econometrics and its Applications August 14, 2012 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
2 Motivation Cannot access p (ω j y o, A) analytically ω j Instead, simulate ω (1) s p y o, A Approximate posterior moments E [h (ω) j y o, A] Bayes actions ba = arg min a2a E [L (a, ω) j y o, A] ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
3 Motivation Cannot access p (ω j y o, A) analytically ω j Instead, simulate ω (1) s p y o, A Approximate posterior moments E [h (ω) j y o, A] Bayes actions ba = arg min a2a E [L (a, ω) j y o, A] ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
4 Notation The methods we discuss today can be used for a probability distribution of interest, I To this end: θ (m) sp (θ ji ) ω (m) sp ω j θ (m),i ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
5 Notation The methods we discuss today can be used for a probability distribution of interest, I To this end: θ (m) sp (θ ji ) ω (m) sp ω j θ (m),i ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
6 The setup Suppose the following is possible: θ (m) iid sp (θ ji ) ω (m)iid sp ω j θ (m),i Our setup (Theorem 4.1.1) n θ (m), ω (m)o is i.i.d.; θ (m) 2 Θ, ω (m) 2 Ω, θ (m) s p (θ j I ) h : Ω! R 1 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
7 The setup Suppose the following is possible: θ (m) iid sp (θ ji ) ω (m)iid sp ω j θ (m),i Our setup (Theorem 4.1.1) n θ (m), ω (m)o is i.i.d.; θ (m) 2 Θ, ω (m) 2 Ω, θ (m) s p (θ j I ) h : Ω! R 1 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
8 Some conditions First momdent condition: E [h (ω) j I ] = h; Second moment condition: var [h (ω) j I ] = σ 2 ; Probability mass condition: For given p 2 (0, 1), there is a unique h p such that the statements P [h (ω) h p j I ] p and P [h (ω) h p j I ] 1 p are both true; Positive p.d.f. condition: For the unique h p corresponding to p, p [h (ω) = h p j I ] > 0. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
9 Some conditions First momdent condition: E [h (ω) j I ] = h; Second moment condition: var [h (ω) j I ] = σ 2 ; Probability mass condition: For given p 2 (0, 1), there is a unique h p such that the statements P [h (ω) h p j I ] p and P [h (ω) h p j I ] 1 p are both true; Positive p.d.f. condition: For the unique h p corresponding to p, p [h (ω) = h p j I ] > 0. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
10 Given the setup, First result (a) ω (m) i.i.d. s p (ω j I ) whether or not any of conditions 1-4 are true. Given the rst moment condition, h (M ) = M 1 Second result (b) M h ω (m) a.s.! h. m=1 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
11 Given the setup, First result (a) ω (m) i.i.d. s p (ω j I ) whether or not any of conditions 1-4 are true. Given the rst moment condition, h (M ) = M 1 Second result (b) M h ω (m) a.s.! h. m=1 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
12 Third result (c) Given the rst and second moment conditions, M 1/2 h (M ) d! h N 0, σ 2 and bσ 2(M ) = M 1 M h h ω (m) m=1 h (M )i 2 a.s.! σ 2 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
13 Third result (c) Given the rst and second moment conditions, M 1/2 h (M ) d! h N 0, σ 2 and bσ 2(M ) = M 1 M h h ω (m) m=1 h (M )i 2 a.s.! σ 2 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
14 Notation for quantiles Let bh (M ) p be any real number such that M 1 M m=1 h I i,bh p (M ) h ω (m)i p and M 1 M h I h bh p (M ) h ω (m)i 1 p., m=1 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
15 Notation for quantiles Let bh (M ) p be any real number such that M 1 M m=1 h I i,bh p (M ) h ω (m)i p and M 1 M h I h bh p (M ) h ω (m)i 1 p., m=1 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
16 Fourth result (d) Given the probability mass condition, bh (M ) p a.s.! h p Fifth result (e) Given the probabilty mass and positive p.d.f. conditions, N h i M 1/2 bh p (M ) d! h p n 0, p (1 p) /p [h (ω) = h p j I ] 2o. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
17 Fourth result (d) Given the probability mass condition, bh (M ) p a.s.! h p Fifth result (e) Given the probabilty mass and positive p.d.f. conditions, N h i M 1/2 bh p (M ) d! h p n 0, p (1 p) /p [h (ω) = h p j I ] 2o. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
18 Assessment of approximation error Recall the third result (c): M 1/2 h (M ) d! h N 0, σ 2 and bσ 2(M ) = M 1 M h h ω (m) m=1 h (M )i 2 a.s.! σ 2 The numerical standard error of h (M ) is bσ 2(M ) /M 1/2 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
19 Assessment of approximation error Recall the third result (c): M 1/2 h (M ) d! h N 0, σ 2 and bσ 2(M ) = M 1 M h h ω (m) m=1 h (M )i 2 a.s.! σ 2 The numerical standard error of h (M ) is bσ 2(M ) /M 1/2 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
20 Approximation of Bayes actions by direct sampling: The setup θ (m) s p (θ ji ), ω (m) j θ (m), I s p ω j θ (m), I. L (a, ω) 0 is a loss function de ned on Ω A, where A is an open subset of R q. The risk function Z R (a) = Ω Z Θ L (a, ω) p (θ ji ) p (ω j θ, I ) dθdω has a strict global minimum at ba 2 A R m. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
21 Approximation of Bayes actions by direct sampling: The setup θ (m) s p (θ ji ), ω (m) j θ (m), I s p ω j θ (m), I. L (a, ω) 0 is a loss function de ned on Ω A, where A is an open subset of R q. The risk function Z R (a) = Ω Z Θ L (a, ω) p (θ ji ) p (ω j θ, I ) dθdω has a strict global minimum at ba 2 A R m. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
22 First result Under weak regularity conditions (Theorem 4.1.2, conditions (1-2) lim P inf (a ba) 0 (a ba) > ε j I = 0. M! a2a M ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
23 First result Under weak regularity conditions (Theorem 4.1.2, conditions (1-2) lim P inf (a ba) 0 (a ba) > ε j I = 0. M! a2a M ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
24 Second result This requires Conditions 1-6, which include B =var [ L (a, ω) / aj a=ba j I ] exists and is nite; H =E 2 L (a, ω) / a a 0 j a=ba j I exists and is nite and nonsingular. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
25 Second result This requires Conditions 1-6, which include B =var [ L (a, ω) / aj a=ba j I ] exists and is nite; H =E 2 L (a, ω) / a a 0 j a=ba j I exists and is nite and nonsingular. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
26 Second result (continued) Then M 1/2 (ba M ba)! d N 0, H 1 BH 1, M 1 M m=1 L a, ω (m) / aj a=bam L a, ω (m) / a 0 p j a=bam! B, M 1 M m=1 2 L a, ω (m) / a a 0 p j a=bam! H. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
27 Second result (continued) Then M 1/2 (ba M ba)! d N 0, H 1 BH 1, M 1 M m=1 L a, ω (m) / aj a=bam L a, ω (m) / a 0 p j a=bam! B, M 1 M m=1 2 L a, ω (m) / a a 0 p j a=bam! H. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
28 Importance sampling: Motivation Suppose that we cannot derive a method for drawing θ (m) iid s p (θ ji ) but we can simulate θ (m) iid s p (θ js) where p (θ js) is similar to p (θ ji ). ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
29 Importance sampling: Motivation Suppose that we cannot derive a method for drawing θ (m) iid s p (θ ji ) but we can simulate θ (m) iid s p (θ js) where p (θ js) is similar to p (θ ji ). ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
30 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
31 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
32 Main results: The setup n u θ (m), ω (m)o is independent and identically distributed, θ (m) s p (θ js) ω (m) s p ω j θ (m), I. De ne the weighting function As before,h : Ω! R 1 w (θ) = p (θ j I ) /p (θ j S) ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
33 Main results: The setup n u θ (m), ω (m)o is independent and identically distributed, θ (m) s p (θ js) ω (m) s p ω j θ (m), I. De ne the weighting function As before,h : Ω! R 1 w (θ) = p (θ j I ) /p (θ j S) ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
34 Some conditions First moment condition: E [h (ω) j I ] = h exists Second moment condition: var [h (ω) j I ] = σ 2 exists Essential condition: The support of p (θ js) includes Θ. Bounded weight condtion: w (θ) = p (θ j I ) /p (θ j S) is bounded above on Θ. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
35 Some conditions First moment condition: E [h (ω) j I ] = h exists Second moment condition: var [h (ω) j I ] = σ 2 exists Essential condition: The support of p (θ js) includes Θ. Bounded weight condtion: w (θ) = p (θ j I ) /p (θ j S) is bounded above on Θ. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
36 More notation Kernels k (θ ji ) = c I p (θ ji ) k (θ js) = c S p (θ js) ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
37 More notation Kernels k (θ ji ) = c I p (θ ji ) k (θ js) = c S p (θ js) ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
38 First result Given the rst moment condition and the essential condition, h (M ) = M m=1 w θ (m) h ω (m) a.s. M m=1 w θ (m)! h The proof is straightforward. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
39 First result Given the rst moment condition and the essential condition, h (M ) = M m=1 w θ (m) h ω (m) a.s. M m=1 w θ (m)! h The proof is straightforward. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
40 Second result Given all four conditons, M 1/2 h (M ) d! h N 0, τ 2 and h bτ 2(M ) = M M m=1 h ω (m) h (M )i 2 w θ (m) 2 h M m=1 w θ (m)i 2 a.s.! τ 2. The essential part of the proof is the delta method. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
41 Second result Given all four conditons, M 1/2 h (M ) d! h N 0, τ 2 and h bτ 2(M ) = M M m=1 h ω (m) h (M )i 2 w θ (m) 2 h M m=1 w θ (m)i 2 a.s.! τ 2. The essential part of the proof is the delta method. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
42 Bayes actions These results can be extended to provide simulation approximations of the Bayes action ba = arg min E [l (ω) j y o, A] and the Bayes risk Z R (a) = (Theorem 4.2.3) Ω Z Θ L (a, ω) p (θ ji ) p (ω j θ, I ) dθdω. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
43 Bayes actions These results can be extended to provide simulation approximations of the Bayes action ba = arg min E [l (ω) j y o, A] and the Bayes risk Z R (a) = (Theorem 4.2.3) Ω Z Θ L (a, ω) p (θ ji ) p (ω j θ, I ) dθdω. ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
44 A particularly useful property of importance sampling Recall the marginal likelihood Z p (y o j A) = p (θ A j A) p (y o j θ A, A) dθ A Θ A Z p (θ = A j A) p (y o j θ A, A) p (θ A j S) dθ A Θ A p (θ A j S) p (θa j A) p (y = o j θ A, A) E p (θ A j S) if θ A s p (θ A j S). ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
45 A particularly useful property of importance sampling Recall the marginal likelihood Z p (y o j A) = p (θ A j A) p (y o j θ A, A) dθ A Θ A Z p (θ = A j A) p (y o j θ A, A) p (θ A j S) dθ A Θ A p (θ A j S) p (θa j A) p (y = o j θ A, A) E p (θ A j S) if θ A s p (θ A j S). ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
46 If θ (m) A Useful property (continued) p p (y o (θa j A) p (y j A) = o j θ A, A) E p (θ A j S) iid s p (θa j S) then = E 2 E 4 1 M " 1 M M p θ (m) A j A p m=1 M w m=1 θ (m) A p (θ m A j S) #. y o j θ (m) A, A 3 5 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
47 If θ (m) A Useful property (continued) p p (y o (θa j A) p (y j A) = o j θ A, A) E p (θ A j S) iid s p (θa j S) then = E 2 E 4 1 M " 1 M M p θ (m) A j A p m=1 M w m=1 θ (m) A p (θ m A j S) #. y o j θ (m) A, A 3 5 ohn Geweke Bayesian Econometrics and its Applications Session 2B: Some () basic simulation methods August 14, / 24
Session 3A: Markov chain Monte Carlo (MCMC)
Session 3A: Markov chain Monte Carlo (MCMC) John Geweke Bayesian Econometrics and its Applications August 15, 2012 ohn Geweke Bayesian Econometrics and its Session Applications 3A: Markov () chain Monte
More informationCombining Macroeconomic Models for Prediction
Combining Macroeconomic Models for Prediction John Geweke University of Technology Sydney 15th Australasian Macro Workshop April 8, 2010 Outline 1 Optimal prediction pools 2 Models and data 3 Optimal pools
More informationModeling conditional distributions with mixture models: Applications in finance and financial decision-making
Modeling conditional distributions with mixture models: Applications in finance and financial decision-making John Geweke University of Iowa, USA Journal of Applied Econometrics Invited Lecture Università
More informationSession 5B: A worked example EGARCH model
Session 5B: A worked example EGARCH model John Geweke Bayesian Econometrics and its Applications August 7, worked example EGARCH model August 7, / 6 EGARCH Exponential generalized autoregressive conditional
More informationMarkov Chain Monte Carlo Methods
Markov Chain Monte Carlo Methods John Geweke University of Iowa, USA 2005 Institute on Computational Economics University of Chicago - Argonne National Laboaratories July 22, 2005 The problem p (θ, ω I)
More informationBayesian Modeling of Conditional Distributions
Bayesian Modeling of Conditional Distributions John Geweke University of Iowa Indiana University Department of Economics February 27, 2007 Outline Motivation Model description Methods of inference Earnings
More informationDavid Giles Bayesian Econometrics
David Giles Bayesian Econometrics 1. General Background 2. Constructing Prior Distributions 3. Properties of Bayes Estimators and Tests 4. Bayesian Analysis of the Multiple Regression Model 5. Bayesian
More informationBayesian Semiparametric GARCH Models
Bayesian Semiparametric GARCH Models Xibin (Bill) Zhang and Maxwell L. King Department of Econometrics and Business Statistics Faculty of Business and Economics xibin.zhang@monash.edu Quantitative Methods
More informationMODEL COMPARISON CHRISTOPHER A. SIMS PRINCETON UNIVERSITY
ECO 513 Fall 2008 MODEL COMPARISON CHRISTOPHER A. SIMS PRINCETON UNIVERSITY SIMS@PRINCETON.EDU 1. MODEL COMPARISON AS ESTIMATING A DISCRETE PARAMETER Data Y, models 1 and 2, parameter vectors θ 1, θ 2.
More informationCS 540: Machine Learning Lecture 2: Review of Probability & Statistics
CS 540: Machine Learning Lecture 2: Review of Probability & Statistics AD January 2008 AD () January 2008 1 / 35 Outline Probability theory (PRML, Section 1.2) Statistics (PRML, Sections 2.1-2.4) AD ()
More informationE cient Importance Sampling
E cient David N. DeJong University of Pittsburgh Spring 2008 Our goal is to calculate integrals of the form Z G (Y ) = ϕ (θ; Y ) dθ. Special case (e.g., posterior moment): Z G (Y ) = Θ Θ φ (θ; Y ) p (θjy
More informationLecture 7 October 13
STATS 300A: Theory of Statistics Fall 2015 Lecture 7 October 13 Lecturer: Lester Mackey Scribe: Jing Miao and Xiuyuan Lu 7.1 Recap So far, we have investigated various criteria for optimal inference. We
More informationBayesian Semiparametric GARCH Models
Bayesian Semiparametric GARCH Models Xibin (Bill) Zhang and Maxwell L. King Department of Econometrics and Business Statistics Faculty of Business and Economics xibin.zhang@monash.edu Quantitative Methods
More informationStatistics: Learning models from data
DS-GA 1002 Lecture notes 5 October 19, 2015 Statistics: Learning models from data Learning models from data that are assumed to be generated probabilistically from a certain unknown distribution is a crucial
More informationStatistical Approaches to Learning and Discovery. Week 4: Decision Theory and Risk Minimization. February 3, 2003
Statistical Approaches to Learning and Discovery Week 4: Decision Theory and Risk Minimization February 3, 2003 Recall From Last Time Bayesian expected loss is ρ(π, a) = E π [L(θ, a)] = L(θ, a) df π (θ)
More informationModel comparison. Christopher A. Sims Princeton University October 18, 2016
ECO 513 Fall 2008 Model comparison Christopher A. Sims Princeton University sims@princeton.edu October 18, 2016 c 2016 by Christopher A. Sims. This document may be reproduced for educational and research
More informationDavid Giles Bayesian Econometrics
9. Model Selection - Theory David Giles Bayesian Econometrics One nice feature of the Bayesian analysis is that we can apply it to drawing inferences about entire models, not just parameters. Can't do
More informationParametric Models. Dr. Shuang LIANG. School of Software Engineering TongJi University Fall, 2012
Parametric Models Dr. Shuang LIANG School of Software Engineering TongJi University Fall, 2012 Today s Topics Maximum Likelihood Estimation Bayesian Density Estimation Today s Topics Maximum Likelihood
More informationApproximate Bayesian computation for spatial extremes via open-faced sandwich adjustment
Approximate Bayesian computation for spatial extremes via open-faced sandwich adjustment Ben Shaby SAMSI August 3, 2010 Ben Shaby (SAMSI) OFS adjustment August 3, 2010 1 / 29 Outline 1 Introduction 2 Spatial
More informationGAUSSIAN PROCESS REGRESSION
GAUSSIAN PROCESS REGRESSION CSE 515T Spring 2015 1. BACKGROUND The kernel trick again... The Kernel Trick Consider again the linear regression model: y(x) = φ(x) w + ε, with prior p(w) = N (w; 0, Σ). The
More informationLecture 8: Information Theory and Statistics
Lecture 8: Information Theory and Statistics Part II: Hypothesis Testing and I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw December 23, 2015 1 / 50 I-Hsiang
More informationBayesian Inference and MCMC
Bayesian Inference and MCMC Aryan Arbabi Partly based on MCMC slides from CSC412 Fall 2018 1 / 18 Bayesian Inference - Motivation Consider we have a data set D = {x 1,..., x n }. E.g each x i can be the
More informationIntroduction to Systems Analysis and Decision Making Prepared by: Jakub Tomczak
Introduction to Systems Analysis and Decision Making Prepared by: Jakub Tomczak 1 Introduction. Random variables During the course we are interested in reasoning about considered phenomenon. In other words,
More informationClassical and Bayesian inference
Classical and Bayesian inference AMS 132 Claudia Wehrhahn (UCSC) Classical and Bayesian inference January 8 1 / 11 The Prior Distribution Definition Suppose that one has a statistical model with parameter
More informationTheory of Maximum Likelihood Estimation. Konstantin Kashin
Gov 2001 Section 5: Theory of Maximum Likelihood Estimation Konstantin Kashin February 28, 2013 Outline Introduction Likelihood Examples of MLE Variance of MLE Asymptotic Properties What is Statistical
More informationRecitation 2: Probability
Recitation 2: Probability Colin White, Kenny Marino January 23, 2018 Outline Facts about sets Definitions and facts about probability Random Variables and Joint Distributions Characteristics of distributions
More informationPrediction Using Several Macroeconomic Models
Gianni Amisano European Central Bank 1 John Geweke University of Technology Sydney, Erasmus University, and University of Colorado European Central Bank May 5, 2012 1 The views expressed do not represent
More informationAn Introduction to Bayesian Linear Regression
An Introduction to Bayesian Linear Regression APPM 5720: Bayesian Computation Fall 2018 A SIMPLE LINEAR MODEL Suppose that we observe explanatory variables x 1, x 2,..., x n and dependent variables y 1,
More informationModeling conditional distributions with mixture models: Theory and Inference
Modeling conditional distributions with mixture models: Theory and Inference John Geweke University of Iowa, USA Journal of Applied Econometrics Invited Lecture Università di Venezia Italia June 2, 2005
More informationCSC321 Lecture 18: Learning Probabilistic Models
CSC321 Lecture 18: Learning Probabilistic Models Roger Grosse Roger Grosse CSC321 Lecture 18: Learning Probabilistic Models 1 / 25 Overview So far in this course: mainly supervised learning Language modeling
More informationBayesian Inference for DSGE Models. Lawrence J. Christiano
Bayesian Inference for DSGE Models Lawrence J. Christiano Outline State space-observer form. convenient for model estimation and many other things. Bayesian inference Bayes rule. Monte Carlo integation.
More informationThe Normal Linear Regression Model with Natural Conjugate Prior. March 7, 2016
The Normal Linear Regression Model with Natural Conjugate Prior March 7, 2016 The Normal Linear Regression Model with Natural Conjugate Prior The plan Estimate simple regression model using Bayesian methods
More informationSTAT J535: Introduction
David B. Hitchcock E-Mail: hitchcock@stat.sc.edu Spring 2012 Chapter 1: Introduction to Bayesian Data Analysis Bayesian statistical inference uses Bayes Law (Bayes Theorem) to combine prior information
More informationA Very Brief Summary of Statistical Inference, and Examples
A Very Brief Summary of Statistical Inference, and Examples Trinity Term 2008 Prof. Gesine Reinert 1 Data x = x 1, x 2,..., x n, realisations of random variables X 1, X 2,..., X n with distribution (model)
More information2 Statistical Estimation: Basic Concepts
Technion Israel Institute of Technology, Department of Electrical Engineering Estimation and Identification in Dynamical Systems (048825) Lecture Notes, Fall 2009, Prof. N. Shimkin 2 Statistical Estimation:
More informationChapter 9: Interval Estimation and Confidence Sets Lecture 16: Confidence sets and credible sets
Chapter 9: Interval Estimation and Confidence Sets Lecture 16: Confidence sets and credible sets Confidence sets We consider a sample X from a population indexed by θ Θ R k. We are interested in ϑ, a vector-valued
More informationMinimax Estimation of Kernel Mean Embeddings
Minimax Estimation of Kernel Mean Embeddings Bharath K. Sriperumbudur Department of Statistics Pennsylvania State University Gatsby Computational Neuroscience Unit May 4, 2016 Collaborators Dr. Ilya Tolstikhin
More information... Econometric Methods for the Analysis of Dynamic General Equilibrium Models
... Econometric Methods for the Analysis of Dynamic General Equilibrium Models 1 Overview Multiple Equation Methods State space-observer form Three Examples of Versatility of state space-observer form:
More information1 Hypothesis Testing and Model Selection
A Short Course on Bayesian Inference (based on An Introduction to Bayesian Analysis: Theory and Methods by Ghosh, Delampady and Samanta) Module 6: From Chapter 6 of GDS 1 Hypothesis Testing and Model Selection
More informationSolutions of the Financial Risk Management Examination
Solutions of the Financial Risk Management Examination Thierry Roncalli January 9 th 03 Remark The first five questions are corrected in TR-GDR and in the document of exercise solutions, which is available
More informationMAS3301 Bayesian Statistics
MAS3301 Bayesian Statistics M. Farrow School of Mathematics and Statistics Newcastle University Semester 2, 2008-9 1 11 Conjugate Priors IV: The Dirichlet distribution and multinomial observations 11.1
More informationSTAT215: Solutions for Homework 2
STAT25: Solutions for Homework 2 Due: Wednesday, Feb 4. (0 pt) Suppose we take one observation, X, from the discrete distribution, x 2 0 2 Pr(X x θ) ( θ)/4 θ/2 /2 (3 θ)/2 θ/4, 0 θ Find an unbiased estimator
More informationSome Asymptotic Bayesian Inference (background to Chapter 2 of Tanner s book)
Some Asymptotic Bayesian Inference (background to Chapter 2 of Tanner s book) Principal Topics The approach to certainty with increasing evidence. The approach to consensus for several agents, with increasing
More informationGMM-based inference in the AR(1) panel data model for parameter values where local identi cation fails
GMM-based inference in the AR() panel data model for parameter values where local identi cation fails Edith Madsen entre for Applied Microeconometrics (AM) Department of Economics, University of openhagen,
More informationBayesian Inference for DSGE Models. Lawrence J. Christiano
Bayesian Inference for DSGE Models Lawrence J. Christiano Outline State space-observer form. convenient for model estimation and many other things. Preliminaries. Probabilities. Maximum Likelihood. Bayesian
More informationEco517 Fall 2014 C. Sims MIDTERM EXAM
Eco57 Fall 204 C. Sims MIDTERM EXAM You have 90 minutes for this exam and there are a total of 90 points. The points for each question are listed at the beginning of the question. Answer all questions.
More informationIntroduction to Bayesian Inference
University of Pennsylvania EABCN Training School May 10, 2016 Bayesian Inference Ingredients of Bayesian Analysis: Likelihood function p(y φ) Prior density p(φ) Marginal data density p(y ) = p(y φ)p(φ)dφ
More informationIntroduction to Probabilistic Machine Learning
Introduction to Probabilistic Machine Learning Piyush Rai Dept. of CSE, IIT Kanpur (Mini-course 1) Nov 03, 2015 Piyush Rai (IIT Kanpur) Introduction to Probabilistic Machine Learning 1 Machine Learning
More informationMassachusetts Institute of Technology Department of Economics Time Series Lecture 6: Additional Results for VAR s
Massachusetts Institute of Technology Department of Economics Time Series 14.384 Guido Kuersteiner Lecture 6: Additional Results for VAR s 6.1. Confidence Intervals for Impulse Response Functions There
More informationModule 22: Bayesian Methods Lecture 9 A: Default prior selection
Module 22: Bayesian Methods Lecture 9 A: Default prior selection Peter Hoff Departments of Statistics and Biostatistics University of Washington Outline Jeffreys prior Unit information priors Empirical
More informationIntroduction to Probability and Statistics (Continued)
Introduction to Probability and Statistics (Continued) Prof. icholas Zabaras Center for Informatics and Computational Science https://cics.nd.edu/ University of otre Dame otre Dame, Indiana, USA Email:
More informationStatistical Inference: Maximum Likelihood and Bayesian Approaches
Statistical Inference: Maximum Likelihood and Bayesian Approaches Surya Tokdar From model to inference So a statistical analysis begins by setting up a model {f (x θ) : θ Θ} for data X. Next we observe
More informationGaussian kernel GARCH models
Gaussian kernel GARCH models Xibin (Bill) Zhang and Maxwell L. King Department of Econometrics and Business Statistics Faculty of Business and Economics 7 June 2013 Motivation A regression model is often
More informationShort Questions (Do two out of three) 15 points each
Econometrics Short Questions Do two out of three) 5 points each ) Let y = Xβ + u and Z be a set of instruments for X When we estimate β with OLS we project y onto the space spanned by X along a path orthogonal
More informationInverse Statistical Learning
Inverse Statistical Learning Minimax theory, adaptation and algorithm avec (par ordre d apparition) C. Marteau, M. Chichignoud, C. Brunet and S. Souchet Dijon, le 15 janvier 2014 Inverse Statistical Learning
More informationIntroduction to Machine Learning. Lecture 2
Introduction to Machine Learning Lecturer: Eran Halperin Lecture 2 Fall Semester Scribe: Yishay Mansour Some of the material was not presented in class (and is marked with a side line) and is given for
More informationDavid Giles Bayesian Econometrics
David Giles Bayesian Econometrics 5. Bayesian Computation Historically, the computational "cost" of Bayesian methods greatly limited their application. For instance, by Bayes' Theorem: p(θ y) = p(θ)p(y
More informationLikelihood-free MCMC
Bayesian inference for stable distributions with applications in finance Department of Mathematics University of Leicester September 2, 2011 MSc project final presentation Outline 1 2 3 4 Classical Monte
More informationBayesian and Frequentist Methods in Bandit Models
Bayesian and Frequentist Methods in Bandit Models Emilie Kaufmann, Telecom ParisTech Bayes In Paris, ENSAE, October 24th, 2013 Emilie Kaufmann (Telecom ParisTech) Bayesian and Frequentist Bandits BIP,
More informationLecture 8: Information Theory and Statistics
Lecture 8: Information Theory and Statistics Part II: Hypothesis Testing and Estimation I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw December 22, 2015
More informationAsymptotics of minimax stochastic programs
Asymptotics of minimax stochastic programs Alexander Shapiro Abstract. We discuss in this paper asymptotics of the sample average approximation (SAA) of the optimal value of a minimax stochastic programming
More informationVariational Bayes. A key quantity in Bayesian inference is the marginal likelihood of a set of data D given a model M
A key quantity in Bayesian inference is the marginal likelihood of a set of data D given a model M PD M = PD θ, MPθ Mdθ Lecture 14 : Variational Bayes where θ are the parameters of the model and Pθ M is
More informationLecture 1: Bayesian Framework Basics
Lecture 1: Bayesian Framework Basics Melih Kandemir melih.kandemir@iwr.uni-heidelberg.de April 21, 2014 What is this course about? Building Bayesian machine learning models Performing the inference of
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 informationMIT Spring 2016
MIT 18.655 Dr. Kempthorne Spring 2016 1 MIT 18.655 Outline 1 2 MIT 18.655 Decision Problem: Basic Components P = {P θ : θ Θ} : parametric model. Θ = {θ}: Parameter space. A{a} : Action space. L(θ, a) :
More informationI. Bayesian econometrics
I. Bayesian econometrics A. Introduction B. Bayesian inference in the univariate regression model C. Statistical decision theory D. Large sample results E. Diffuse priors F. Numerical Bayesian methods
More information3.0.1 Multivariate version and tensor product of experiments
ECE598: Information-theoretic methods in high-dimensional statistics Spring 2016 Lecture 3: Minimax risk of GLM and four extensions Lecturer: Yihong Wu Scribe: Ashok Vardhan, Jan 28, 2016 [Ed. Mar 24]
More informationMIT Spring 2016
MIT 18.655 Dr. Kempthorne Spring 2016 1 MIT 18.655 Outline 1 2 MIT 18.655 3 Decision Problem: Basic Components P = {P θ : θ Θ} : parametric model. Θ = {θ}: Parameter space. A{a} : Action space. L(θ, a)
More informationDecision theory. 1 We may also consider randomized decision rules, where δ maps observed data D to a probability distribution over
Point estimation Suppose we are interested in the value of a parameter θ, for example the unknown bias of a coin. We have already seen how one may use the Bayesian method to reason about θ; namely, we
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 informationBayesian estimation of bandwidths for a nonparametric regression model with a flexible error density
ISSN 1440-771X Australia Department of Econometrics and Business Statistics http://www.buseco.monash.edu.au/depts/ebs/pubs/wpapers/ Bayesian estimation of bandwidths for a nonparametric regression model
More information9 Bayesian inference. 9.1 Subjective probability
9 Bayesian inference 1702-1761 9.1 Subjective probability This is probability regarded as degree of belief. A subjective probability of an event A is assessed as p if you are prepared to stake pm to win
More informationChapter 9: Hypothesis Testing Sections
Chapter 9: Hypothesis Testing Sections 9.1 Problems of Testing Hypotheses 9.2 Testing Simple Hypotheses 9.3 Uniformly Most Powerful Tests Skip: 9.4 Two-Sided Alternatives 9.6 Comparing the Means of Two
More informationLearning Theory. Ingo Steinwart University of Stuttgart. September 4, 2013
Learning Theory Ingo Steinwart University of Stuttgart September 4, 2013 Ingo Steinwart University of Stuttgart () Learning Theory September 4, 2013 1 / 62 Basics Informal Introduction Informal Description
More informationBayesian Indirect Inference and the ABC of GMM
Bayesian Indirect Inference and the ABC of GMM Michael Creel, Jiti Gao, Han Hong, Dennis Kristensen Universitat Autónoma, Barcelona Graduate School of Economics, and MOVE Monash University Stanford University
More informationSTAT 499/962 Topics in Statistics Bayesian Inference and Decision Theory Jan 2018, Handout 01
STAT 499/962 Topics in Statistics Bayesian Inference and Decision Theory Jan 2018, Handout 01 Nasser Sadeghkhani a.sadeghkhani@queensu.ca There are two main schools to statistical inference: 1-frequentist
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@uni-muenster.de Summer
More informationUsing Simulation Methods for Bayesian Econometric Models: Inference, Development, and Communication
Federal Reserve Bank of Minneapolis Research Department Staff Report 49 June 998 Using Simulation Methods for Bayesian Econometric Models: Inference, Development, and Communication John Geweke Federal
More informationPeter Hoff Minimax estimation October 31, Motivation and definition. 2 Least favorable prior 3. 3 Least favorable prior sequence 11
Contents 1 Motivation and definition 1 2 Least favorable prior 3 3 Least favorable prior sequence 11 4 Nonparametric problems 15 5 Minimax and admissibility 18 6 Superefficiency and sparsity 19 Most of
More informationGibbs Sampling in Linear Models #2
Gibbs Sampling in Linear Models #2 Econ 690 Purdue University Outline 1 Linear Regression Model with a Changepoint Example with Temperature Data 2 The Seemingly Unrelated Regressions Model 3 Gibbs sampling
More informationMinimum Message Length Analysis of the Behrens Fisher Problem
Analysis of the Behrens Fisher Problem Enes Makalic and Daniel F Schmidt Centre for MEGA Epidemiology The University of Melbourne Solomonoff 85th Memorial Conference, 2011 Outline Introduction 1 Introduction
More informationGaussian processes. Chuong B. Do (updated by Honglak Lee) November 22, 2008
Gaussian processes Chuong B Do (updated by Honglak Lee) November 22, 2008 Many of the classical machine learning algorithms that we talked about during the first half of this course fit the following pattern:
More informationGeneral Bayesian Inference I
General Bayesian Inference I Outline: Basic concepts, One-parameter models, Noninformative priors. Reading: Chapters 10 and 11 in Kay-I. (Occasional) Simplified Notation. When there is no potential for
More informationBayesian Estimation of DSGE Models 1 Chapter 3: A Crash Course in Bayesian Inference
1 The views expressed in this paper are those of the authors and do not necessarily reflect the views of the Federal Reserve Board of Governors or the Federal Reserve System. Bayesian Estimation of DSGE
More informationEco517 Fall 2004 C. Sims MIDTERM EXAM
Eco517 Fall 2004 C. Sims MIDTERM EXAM Answer all four questions. Each is worth 23 points. Do not devote disproportionate time to any one question unless you have answered all the others. (1) We are considering
More informationEstimation Tasks. Short Course on Image Quality. Matthew A. Kupinski. Introduction
Estimation Tasks Short Course on Image Quality Matthew A. Kupinski Introduction Section 13.3 in B&M Keep in mind the similarities between estimation and classification Image-quality is a statistical concept
More informationPrinciples of Bayesian Inference
Principles of Bayesian Inference 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 informationBayesian semiparametric GARCH models
ISSN 1440-771X 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 informationFoundations of Statistical Inference
Foundations of Statistical Inference Julien Berestycki Department of Statistics University of Oxford MT 2016 Julien Berestycki (University of Oxford) SB2a MT 2016 1 / 20 Lecture 6 : Bayesian Inference
More informationPeter Hoff Minimax estimation November 12, Motivation and definition. 2 Least favorable prior 3. 3 Least favorable prior sequence 11
Contents 1 Motivation and definition 1 2 Least favorable prior 3 3 Least favorable prior sequence 11 4 Nonparametric problems 15 5 Minimax and admissibility 18 6 Superefficiency and sparsity 19 Most of
More informationg-priors for Linear Regression
Stat60: Bayesian Modeling and Inference Lecture Date: March 15, 010 g-priors for Linear Regression Lecturer: Michael I. Jordan Scribe: Andrew H. Chan 1 Linear regression and g-priors In the last lecture,
More informationDS-GA 1003: Machine Learning and Computational Statistics Homework 7: Bayesian Modeling
DS-GA 1003: Machine Learning and Computational Statistics Homework 7: Bayesian Modeling Due: Tuesday, May 10, 2016, at 6pm (Submit via NYU Classes) Instructions: Your answers to the questions below, including
More information1 of 7 7/16/2009 6:12 AM Virtual Laboratories > 7. Point Estimation > 1 2 3 4 5 6 1. Estimators The Basic Statistical Model As usual, our starting point is a random experiment with an underlying sample
More informationLecture 11 and 12: Basic estimation theory
Lecture ad 2: Basic estimatio theory Sprig 202 - EE 94 Networked estimatio ad cotrol Prof. Kha March 2 202 I. MAXIMUM-LIKELIHOOD ESTIMATORS The maximum likelihood priciple is deceptively simple. Louis
More informationMethods of evaluating estimators and best unbiased estimators Hamid R. Rabiee
Stochastic Processes Methods of evaluating estimators and best unbiased estimators Hamid R. Rabiee 1 Outline Methods of Mean Squared Error Bias and Unbiasedness Best Unbiased Estimators CR-Bound for variance
More information2014 Preliminary Examination
014 reliminary Examination 1) Standard error consistency and test statistic asymptotic normality in linear models Consider the model for the observable data y t ; x T t n Y = X + U; (1) where is a k 1
More informationEstimation theory. Parametric estimation. Properties of estimators. Minimum variance estimator. Cramer-Rao bound. Maximum likelihood estimators
Estimation theory Parametric estimation Properties of estimators Minimum variance estimator Cramer-Rao bound Maximum likelihood estimators Confidence intervals Bayesian estimation 1 Random Variables Let
More informationEstimation of moment-based models with latent variables
Estimation of moment-based models with latent variables work in progress Ra aella Giacomini and Giuseppe Ragusa UCL/Cemmap and UCI/Luiss UPenn, 5/4/2010 Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)Moments
More informationFoundations of Statistical Inference
Foundations of Statistical Inference Julien Berestycki Department of Statistics University of Oxford MT 2016 Julien Berestycki (University of Oxford) SB2a MT 2016 1 / 32 Lecture 14 : Variational Bayes
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 Suggested Projects: www.cs.ubc.ca/~arnaud/projects.html First assignement on the web this afternoon: capture/recapture.
More information