Markov Chain Monte Carlo Simulation Made Simple


 Leslie Parrish
 1 years ago
 Views:
Transcription
1 Markov Chain Monte Carlo Simulation Made Simple Alastair Smith Department of Politics New York University April2,2003 1
2 Markov Chain Monte Carlo (MCMC) simualtion is a powerful technique to perform numerical integration. It can be used to numerically estimate complex economometric models. In this paper I describe the intuition behind the process, show its flexiblity and applicability. I conclude by demonstrating that these methods are often simpler to implement than many common techniques such as MLE. This paper serves as a brief introduction. I do not intend to derive any results or prove any theorems. I beleive that MCMC offers a powerful estimation tool. This paper is designed to remove the mystery surround the process. Not only it extremely powerful and flexible but it is easy to implement. Given the recent growth in the power of computers I beleive that numerical procedures will be the estimation tools of the future. I outline the underlying logic, show why these techniques work. MCMC techniques are most often used in the Bayesian context. I start by outlining the simple linear model in the Bayesian framework. Although, analytical techniques exist for this model they are complex. In general, more complex model are analytically intractable. Having setup the estimation technique I examine the properties of Markov chains. These properties provide the basis for the estimation procedure. 1 The Bayesian Model While I beleive that the Bayesian approach a superior, consistent approach to statistics than the standard frequentist approach, this debate is volumous and not the topic of this paper. For practical purposes if is usually possible to use diffuse priors that do not influence the posterior results. prior f(θ) likelihood L(Y θ) posterior f(θ Y ) f(θ) L(Y θ) For example in the simple linear model θ = {β,σ 2 } 2 Markov Chains A Markov chain is a stochastic process. It generates a series a observations, X. To illustrate the concept I focus on a discrete time, descrete state space model. At each time period the process generates a sample, X t,from 2
3 the state space. For a simple example, suppose that the state space is the numbers 1, 2,and 3. A Markov chain is simply a string of these numbers. The Markov property is that the probability distribution over the next observation depends only upon the current observation. Let p ij represent the probability that the next observation is j (X t+1 = j), given that the current observation is i (X t = i). A convenient way to present these transition probabilities is through a transition matrix P, P = p 11 p 12 p 13 p 21 p 22 p 23. The elements of the first row represent the probabilities of moving to the different states if the current state is 1. Therefore, p 31 p 32 p 33 p 11 represents the probability that state X t+1 =1ifthecurrentstateisalso 1; p 12 represents the probability that state X t+1 =2if the current state is 1, etc... Suppose that our initial observation is indeed 1 (X 0 =1). The probability distribution for state is given by the first row of P. The next question is what is the probability distribution over the following observation. To illustrate, I consider the more specific question, with what probability does X 2 =3? There are three possible paths by which the second observation could equal 3; they are illustrated in the table below. Thus the probability that X 2 =3 is p 11 p 13 + p 12 p 23 + p 13 p 33. Pathway X 0 X 1 X 2 Probability # p 11 p 13 # p 12 p 23 # p 13 p 33 Thus, for any initial state, we can calculate the probability density over the states for a given number of moves. Obviously, as the number of moves increases these become increasingly difficult to calculate. Yet, matrix notation simplifies the calculation. Suppose, rather than start with a specific state we consider a probability distribution over these states, ν 0 = 0. v1 0 v2 0 v3 0 If we randomly select the initial state from this distribution, then what is the probability distribution of the next state in the chain is given by v (1) 1 v (1) = v (1) 2 = Pv (0) = p 11 p 12 p 13 p 21 p 22 p 23 v (1) p 31 p 32 p 33 3 v0 1 v 0 2 v 0 3 = p 11v p 12 v p 13 v 0 3 p 21 v p 22 v p 23 v 0 3 p 31 v p 32 v p 33 v 0 3 3
4 This idea can be extended, the probability distribution over the states after the second move is simply v 2 = Pv 1 = P 2 v 0. This idea can be generalised; specifically, v (t) = P t v (0). Of particular interest, is the distribution as the chain becomes long. As the chain s length increases then the distribution over the states becomes less and less determined by the starting distribution and more and more determined by the transition probabilities. Indeed, providing the chain satisfies certain regularity conditions, i.e. it does not get stuck in one state, there exists a unique invariant distribution associated with every transition matrix. Let π represent this invariant distribution. So for any starting distribution, π (0), as the chain becomes long then the π (t) tends to π ( lim π (t) = π). t There are two ways to calculate this invariant distribution. The first is analytical. This method exploits the fact that π = Pπ, and solves this system of equations. The second, and of more relevance for this paper, is to similate π by actually running the Markov chain. This involves choosing a starting value and simply running the Markov chain. The initial values in the chain depend strongly upon the starting values. However, as the chain becomes longer then the elements of the chain represent random draws from the probability distribution π Suppose, for example, that the transition matrix is P = We could start by setting X 0 =1and then running the Markov chain. We could estimate the density of each state by examining the frequency of each state. Figure 1 demonstrates that, as the number of iterations becomes large, that the relative frequency of each state converges to its invariant density. We can arbitarily increase the accuracy of these estimates simply by taking more iteration. In this example, I use a discrete state space model; however, these ideas are readily extendable to continuous state space models, where the transition matrix is replaced by a transition kernel (a probability density over the next state that depends only upon the current state). 2.1 Exploiting Markov chains for estimation Most of Markov theory revolves around finding the invariant distribution of Markov chains. MCMC turns the problem arround. Rather than finding 4
5 x1 x2 x N Figure 1: 5
6 the invariant distribution of a specific Markov chain, it starts with a specific invariant distributions and says, can I find a Markov chain that has this invariant distribution. 1 Typically, we already know the distribution of interest: the posterior distribution of the parameters. The key is to find a transition kernel that has this invariant distribution f(θ Y ). In Bayesian estimation we want to find the posterior distribution of the parameters, f(θ Y ). As discussed above this is often analytically intractible. However, suppose we have a Markov chain, P, whose invariant distribution is f(θ Y ). If we run thismarkovprocessthen,asthechainbecomeslong,itselementsrepresent random draws from the posterior distribution f(θ Y ). To illustrate how the process works consider the following algorthym. 1. Choose starting values, θ (0), and length of the chain, n 0 + m. 2. Given the current element in the chain, θ (t), use the Markov process P, to draw the next element θ (t+1). 3. If t>m,thenstoreθ (t+1). 4. If t<m+ n 0, then return to step 2; otherwise calculate and report the descriptive statistics for the elements stored in step 3. This algorthym generates and stores n 0 elements from the chain. These elements represent random samples from the posterior distribution of f (θ Y ). Thus the sample average represents an estimate of the expected value of θ. Other properties of f (θ Y ) can also be estimated by examining the properties of the sample. The accuracy of these estimates depends upon the number of draws, n 0. Accurracy is improved by running the chain longer. Note that the first m iterations of the chain were discarded. The initial elements in the chain are strongly influenced by the starting value (as the figure above demonstrates). If these starting values are drawn from a low density region of the posterior denisty then the chain contains too many draws from this region. 2 1 Each Markov process has a unique invariant distribution. Yet, many Markov chains could have the same invariant distribution. Thus, we are free to use any of these process to simulate the invariant distribution. 2 Another practical problem with running this algorthym is the high autocorrelation between elements in the chain. This reduces the rate a which convergence is acheived. A practical solution is to subsample from elements stored at step 3. 6
7 In summary, if we can find a Markov process with transition kernel P, such that its invariant distribution is f (θ Y ), then we can numerically estimate this posterior distribution by running the Markov chain. Obviously, there are many importance convergence consideration that I have not considered. However, the basic point is that if an appropriate transition kernel can be found, then estimations involves nothing more than running the Markov process. So far I have said nothing about how to find an appropriate transition kernel. It is to this point that I turn next. 3 Transition Kernels Table 1 compares the analytically calculated probability distribution with the numerically simulated values. The accuracy of the simulation can be increased by simply increases the number of iterations of the chain. 3 Most of Markov theory revolves around finding the invariant distribution of Markov chains. MCMC turns the problem arround. Typically, we already know the distribution of interest: the posterior distribution of the parameters. The key is to find a transition kernal that has this invariant distribution. Then to estimate this distribution we simply need to run the Markov chain for a suitably long period. 4 Joint, marginal and conditional distributions In the linear model we want to estimate f(β,σ 2 Y ). Being somewhat informal, this is the probability density of seeing a particular value of β and σ 2. Bayesian have calculated this density. It turns out that, with suitable conjugate priors 4, f(β,σ 2 Y ) is distributed inverse gamma normal. Unfortunately, this is about the most complicated model for which we can work with 3 These is a convenient time to discuss several aspects associated with implementing MCMC methods. First the starting value of the Markov chain affect the initial values of the chain. Over time their effect diminishes. However, if the starting values represent very low density portions of the state space then the choice of starting values affects the results. The usual solution is to discard the early part of the chain. This tends to disregard those draws from the chain that are highly dependent upon the starting values. Convergence criticeria??? literature????? 4 what is a conjugate prior? 7
8 the joint posterior density analytically. For more complex models the joint density is simply intractable. Yet, generally our interest is in the marginal denisty of a particular parameter. In particular case of the simple linear model we typically want to know about β and σ 2, separately. For example, this is all we report from a regression model, the distribution of β. This marginal density is simply the joint density of β and σ 2 intregrated across all possible values of σ 2. The key do using MCMC is to stop thinking in terms of calculating things analytically and imagine how you could simulate a single parameter in a model if you knew all the other parameters. Suppose for example that you knew the marginal distribution of σ 2 and wanted to calculate the marginal distribution of β. In order to estimate the marginal density of β Icould simply, integrate out σ 2 from the joint density. While simply tricky in this problem, it is impossible in more complex econometric models. However, knowing the marginal density of σ 2, I can draw a large number of random draws from this density. For each of these draws, the conditional density of β is simple to calculate (with normal priors, f(β Y,σ 2 ) is also distributed normally). To numerically estimate β I could draw a random sample from this distribution. Algorithm to calculate the marginal density of β given that the density of σ 2 is known. 1. set t=1 2. randonly draw (σ 2 ) (t) from its known posterior marginal distribution 3. calculate the posterior density of β given (σ 2 ) (t) (f(β Y,(σ 2 ) (t) )) 4. randomly draw (β) t from f(β Y,σ 2(t) )) 5. let t=t+1 and go to 2 Suppose this algorthm is repeated T times. Then the T samples of β represent random draws from its marginal density. The algorthm effectively integrates out σ 2. As an analogy, in our 101 econometrics classes we learn how to estimate the means of a variable if we know its variance. We then learn to calculate the variance if we know the mean. Being an order of magnitude harder, the calculation of the joint distribution of the mean and variance is typically 8
9 ommitted. Calculating the posterior density of the mean and the variance togther is much harder than calculating either conditional density. However, providing we can break a model down into a series of simple conditional densities we can estimate the marginal density of a parameter. The algorithm above assumed that the distribution of σ 2 was known and it produced a random sample from the posterior density of β. However,ifthe draws from the algorithm represent random draws from the marginal density of β, then we could simply reverse the logic of the argument, and draw random samples from the conditional density of σ 2 given the current value of β. Given that the β s are random draws from the marginal density for β, then random draws of σ 2 represent random draws from the marginal density of σ 2. Hence the following algorithm simulates the posterior ditributions for β and σ 2. Algorithm to calculate the marginal density of β and σ set t=1 and choose starting values, β (0) and (σ 2 ) (0). 2. calculate the posterior density of β given (σ 2 ) (t) (f(β Y,(σ 2 ) (t) )) 3. Randomly β (t+1) draw from this distribution. 4. calculate the posterior density of (σ 2 ) (t) given (β) (t+1) (f((σ 2 ) (t) Y,(β) (t+1) )) 5. randomly draw (σ 2 ) (t+1) from f((σ 2 ) Y,(β) (t+1) ) 6. let t=t+1 and go to 2 Providing the prior are appropriately choosen then the calculates of f(β Y,(σ 2 ) (t) ) and f((σ 2 ) (t) Y,(β) (t+1) ) are straightforward. The following code shows how simply this algorthym can be implied in STATA. See program OLS_MCMC.do 5 Bayesian Updates for simple models. Suppose we assume that the likelihood function is normal and so is our prior: Likelihood: p(y θ) = 1 2π exp ( 1(y 2 θ)2 ) 9
10 Normal prior: f(θ) = 1 2π exp ( 1 2 (θ µ 0) 2 ). To make life as simple as possible, suppose initially that the variance of both the likelihood and the prior density in one. By Bayes rule the posterior density is proportional to the product of the prior and the likelihood: p(θ y) p(y θ)f(θ). We can show that posterior denisty is also normal. Specifically, p(θ y) exp ( 1 2 (y θ)2 )exp ( 1 2 (θ µ 0) 2 )=exp ( 1 2 [(y θ)2 +(θ µ 0 ) 2 ]) We can expand the terms in the exponential and then collect them (completing the square). 1 2 [(y θ)2 +(θ µ 0 ) 2 ]= θ 2 +( y µ 0 ) θ y µ2 0. We only care about term is θ since everything else is in the nomalizing constant. θ 2 +( y µ 0 ) θ y µ2 0 =(θ b) 2 = θ 2 2θb+b 2 so b = 1 (y + µ 2 0) Hence p(θ y) exp ((θ 1 2 (y + µ 0)) 2 ) so p(θ y) is distributed normal with mean 1 2 (y + µ 0) and variance 1 2.We can now move to a more realistic example. The normal prior is referred to as a conjugate prior since it results in posterior density from the same class of distributions. 5.1 Simple Linear Model Consider the OLS simple linear model, y i = x i β + e i where e i N(0,σ 2 ). Using conjugate prior, β 0 N(β 0,B 0 ) and σ 2 0 GAMMA(υ 0,δ 0 ),wecan derive the posterior conditional densities. The posterior β parameter is normally distributed: β y, σ 2 N( β,b) b where β b = B(B0 1 β 0 + P N i=1 x iy i ) and B =(B0 1 + P N i=1 x0 ix i ) 1,and The posterior σ 2 is inverse gamma distributed: So σ 2 is distributed G( υ 0+N, δ 0+SSE ) 2 2 where SSE = P N i=1 (y i x i β) 2 i.e. sum of squared errors. 5.2 More Complex Models A key advantage of MCMC is that models can be built up in simple stepwise fashion. Suppose from example, that instead of a continuous dependent variable we have binary outcomes. Such data is typically analysed as a probit model. Specifically, z i = x i β + e i where e i N(0, 1), asify i =1 then z i > 0 and if y i =0then z i < 0. The variable z i is referred to a latent variable since we never actually observe it. The standard approach 10
11 to estimating such a model is to integrate out the latent variable and then apply maximium likelihood. A simple MCMC approach utilitizes a data augmentation technique (Tanner and Wong, 198?). If we knew the value of these latent data then we could simulate the β sjustaswedidintheols model above. Although we don t care directly about the latent data we can simulate these data. The probit model tells us the distribution of the latent data. Specific, if y i =1then z i has a left truncated normal distribution with mean xβ and variance 1: x TN [0,+ ] (x i β,1). Similarly, if y i =0 then we know that the corresponding latent variable lies between and 0: x TN [,0] (x i β,1). We now that the tools to implement this model. Let Z refer to the set of latent data (i.e. all the z i s.) Algorithm to calculate the marginal density of β in a probit model. 1. set t=1 and choose starting values, β (0) and (Z) (0). 2. calculate the posterior density of β given Z (t) (f(β Z t )) 3. Randomly β (t+1) draw from this distribution. 4. calculate the posterior density of Z (t) given (β) (t+1) (f(z Y,(β) (t+1) )) 5. randomly draw (Z) (t+1) from f(z Y,(β) (t+1) ) 6. let t=t+1 and go to 2 The key simplification here is that given Z, the posterior distribution of β is independent of the binary observed dependent variable. Now while in this context, the MLE approach provides highly reliable estimates in more complex models, such are multivariate, multinominal, or censored descrete choice models, MLE is less reliable. MCMC provides a powerful tool in these cases, being easy to program and less prone to the convergence failure problems of MLE. The construction of MCMC can be done peicewise. For example, the OLS code above with estimate the probit model with two additions. First, set the variance, σ 2, equal to one. Second, add the set to draw the latent data, Z. This is easily acheived using the following simulation. If z is a truncated normal variable with mean xβ, variance 1 with a range p to q, then the following algorithym readily provides a method to genrate a random sample 11
12 from the distribution of z. If x TN [p,q] (µ, σ 2 ) and u is a uniform random number then x = µ + σφ 1 (Φ((p µ)/σ)+u(φ((q µ)/σ) Φ((p µ)/σ))), represents a random draw of x. 12
Bayesian Statistics in One Hour. Patrick Lam
Bayesian Statistics in One Hour Patrick Lam Outline Introduction Bayesian Models Applications Missing Data Hierarchical Models Outline Introduction Bayesian Models Applications Missing Data Hierarchical
More informationBasics of Statistical Machine Learning
CS761 Spring 2013 Advanced Machine Learning Basics of Statistical Machine Learning Lecturer: Xiaojin Zhu jerryzhu@cs.wisc.edu Modern machine learning is rooted in statistics. You will find many familiar
More informationWebbased Supplementary Materials for Bayesian Effect Estimation. Accounting for Adjustment Uncertainty by Chi Wang, Giovanni
1 Webbased Supplementary Materials for Bayesian Effect Estimation Accounting for Adjustment Uncertainty by Chi Wang, Giovanni Parmigiani, and Francesca Dominici In Web Appendix A, we provide detailed
More informationParametric Models Part I: Maximum Likelihood and Bayesian Density Estimation
Parametric Models Part I: Maximum Likelihood and Bayesian Density Estimation Selim Aksoy Department of Computer Engineering Bilkent University saksoy@cs.bilkent.edu.tr CS 551, Fall 2015 CS 551, Fall 2015
More information1 Prior Probability and Posterior Probability
Math 541: Statistical Theory II Bayesian Approach to Parameter Estimation Lecturer: Songfeng Zheng 1 Prior Probability and Posterior Probability Consider now a problem of statistical inference in which
More informationInference on Phasetype Models via MCMC
Inference on Phasetype Models via MCMC with application to networks of repairable redundant systems Louis JM Aslett and Simon P Wilson Trinity College Dublin 28 th June 202 Toy Example : Redundant Repairable
More informationThe Exponential Family
The Exponential Family David M. Blei Columbia University November 3, 2015 Definition A probability density in the exponential family has this form where p.x j / D h.x/ expf > t.x/ a./g; (1) is the natural
More information11. Time series and dynamic linear models
11. Time series and dynamic linear models Objective To introduce the Bayesian approach to the modeling and forecasting of time series. Recommended reading West, M. and Harrison, J. (1997). models, (2 nd
More informationEC 6310: Advanced Econometric Theory
EC 6310: Advanced Econometric Theory July 2008 Slides for Lecture on Bayesian Computation in the Nonlinear Regression Model Gary Koop, University of Strathclyde 1 Summary Readings: Chapter 5 of textbook.
More informationLab 8: Introduction to WinBUGS
40.656 Lab 8 008 Lab 8: Introduction to WinBUGS Goals:. Introduce the concepts of Bayesian data analysis.. Learn the basic syntax of WinBUGS. 3. Learn the basics of using WinBUGS in a simple example. Next
More informationBayesian Methods. 1 The Joint Posterior Distribution
Bayesian Methods Every variable in a linear model is a random variable derived from a distribution function. A fixed factor becomes a random variable with possibly a uniform distribution going from a lower
More informationModels for Count Data With Overdispersion
Models for Count Data With Overdispersion Germán Rodríguez November 6, 2013 Abstract This addendum to the WWS 509 notes covers extrapoisson variation and the negative binomial model, with brief appearances
More informationBonusmalus systems and Markov chains
Bonusmalus systems and Markov chains Dutch car insurance bonusmalus system class % increase new class after # claims 0 1 2 >3 14 30 14 9 5 1 13 32.5 14 8 4 1 12 35 13 8 4 1 11 37.5 12 7 3 1 10 40 11
More informationSpatial Statistics Chapter 3 Basics of areal data and areal data modeling
Spatial Statistics Chapter 3 Basics of areal data and areal data modeling Recall areal data also known as lattice data are data Y (s), s D where D is a discrete index set. This usually corresponds to data
More information1 Sufficient statistics
1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =
More informationA Bootstrap MetropolisHastings Algorithm for Bayesian Analysis of Big Data
A Bootstrap MetropolisHastings Algorithm for Bayesian Analysis of Big Data Faming Liang University of Florida August 9, 2015 Abstract MCMC methods have proven to be a very powerful tool for analyzing
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 6 Three Approaches to Classification Construct
More informationCentre for Central Banking Studies
Centre for Central Banking Studies Technical Handbook No. 4 Applied Bayesian econometrics for central bankers Andrew Blake and Haroon Mumtaz CCBS Technical Handbook No. 4 Applied Bayesian econometrics
More informationLecture 3: Linear methods for classification
Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,
More informationBayesian Statistics: Indian Buffet Process
Bayesian Statistics: Indian Buffet Process Ilker Yildirim Department of Brain and Cognitive Sciences University of Rochester Rochester, NY 14627 August 2012 Reference: Most of the material in this note
More informationHypothesis Testing for Beginners
Hypothesis Testing for Beginners Michele Piffer LSE August, 2011 Michele Piffer (LSE) Hypothesis Testing for Beginners August, 2011 1 / 53 One year ago a friend asked me to put down some easytoread notes
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationLinear Classification. Volker Tresp Summer 2015
Linear Classification Volker Tresp Summer 2015 1 Classification Classification is the central task of pattern recognition Sensors supply information about an object: to which class do the object belong
More informationGeneralized Linear Models. Today: definition of GLM, maximum likelihood estimation. Involves choice of a link function (systematic component)
Generalized Linear Models Last time: definition of exponential family, derivation of mean and variance (memorize) Today: definition of GLM, maximum likelihood estimation Include predictors x i through
More informationLecture 6: The Bayesian Approach
Lecture 6: The Bayesian Approach What Did We Do Up to Now? We are given a model Loglinear model, Markov network, Bayesian network, etc. This model induces a distribution P(X) Learning: estimate a set
More informationEconometrics Simple Linear Regression
Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight
More informationValidation of Software for Bayesian Models using Posterior Quantiles. Samantha R. Cook Andrew Gelman Donald B. Rubin DRAFT
Validation of Software for Bayesian Models using Posterior Quantiles Samantha R. Cook Andrew Gelman Donald B. Rubin DRAFT Abstract We present a simulationbased method designed to establish that software
More informationLogistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression
Logistic Regression Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Logistic Regression Preserve linear classification boundaries. By the Bayes rule: Ĝ(x) = arg max
More informationProbabilistic Models for Big Data. Alex Davies and Roger Frigola University of Cambridge 13th February 2014
Probabilistic Models for Big Data Alex Davies and Roger Frigola University of Cambridge 13th February 2014 The State of Big Data Why probabilistic models for Big Data? 1. If you don t have to worry about
More informationCredit Risk Models: An Overview
Credit Risk Models: An Overview Paul Embrechts, Rüdiger Frey, Alexander McNeil ETH Zürich c 2003 (Embrechts, Frey, McNeil) A. Multivariate Models for Portfolio Credit Risk 1. Modelling Dependent Defaults:
More informationCHAPTER 2 Estimating Probabilities
CHAPTER 2 Estimating Probabilities Machine Learning Copyright c 2016. Tom M. Mitchell. All rights reserved. *DRAFT OF January 24, 2016* *PLEASE DO NOT DISTRIBUTE WITHOUT AUTHOR S PERMISSION* This is a
More informationCHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.
Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,
More informationStatistical Machine Learning from Data
Samy Bengio Statistical Machine Learning from Data 1 Statistical Machine Learning from Data Gaussian Mixture Models Samy Bengio IDIAP Research Institute, Martigny, Switzerland, and Ecole Polytechnique
More informationGaussian Conjugate Prior Cheat Sheet
Gaussian Conjugate Prior Cheat Sheet Tom SF Haines 1 Purpose This document contains notes on how to handle the multivariate Gaussian 1 in a Bayesian setting. It focuses on the conjugate prior, its Bayesian
More information3 Random vectors and multivariate normal distribution
3 Random vectors and multivariate normal distribution As we saw in Chapter 1, a natural way to think about repeated measurement data is as a series of random vectors, one vector corresponding to each unit.
More informationLinear Threshold Units
Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear
More informationStandard errors of marginal effects in the heteroskedastic probit model
Standard errors of marginal effects in the heteroskedastic probit model Thomas Cornelißen Discussion Paper No. 320 August 2005 ISSN: 0949 9962 Abstract In nonlinear regression models, such as the heteroskedastic
More informationPS 271B: Quantitative Methods II. Lecture Notes
PS 271B: Quantitative Methods II Lecture Notes Langche Zeng zeng@ucsd.edu The Empirical Research Process; Fundamental Methodological Issues 2 Theory; Data; Models/model selection; Estimation; Inference.
More informationINDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition)
INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) Abstract Indirect inference is a simulationbased method for estimating the parameters of economic models. Its
More informationBayesian logistic betting strategy against probability forecasting. Akimichi Takemura, Univ. Tokyo. November 12, 2012
Bayesian logistic betting strategy against probability forecasting Akimichi Takemura, Univ. Tokyo (joint with Masayuki Kumon, Jing Li and Kei Takeuchi) November 12, 2012 arxiv:1204.3496. To appear in Stochastic
More informationGenerating Random Numbers Variance Reduction QuasiMonte Carlo. Simulation Methods. Leonid Kogan. MIT, Sloan. 15.450, Fall 2010
Simulation Methods Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Simulation Methods 15.450, Fall 2010 1 / 35 Outline 1 Generating Random Numbers 2 Variance Reduction 3 QuasiMonte
More information1 Teaching notes on GMM 1.
Bent E. Sørensen January 23, 2007 1 Teaching notes on GMM 1. Generalized Method of Moment (GMM) estimation is one of two developments in econometrics in the 80ies that revolutionized empirical work in
More informationImputing Missing Data using SAS
ABSTRACT Paper 32952015 Imputing Missing Data using SAS Christopher Yim, California Polytechnic State University, San Luis Obispo Missing data is an unfortunate reality of statistics. However, there are
More informationPoisson Models for Count Data
Chapter 4 Poisson Models for Count Data In this chapter we study loglinear models for count data under the assumption of a Poisson error structure. These models have many applications, not only to the
More informationInterpretation of Somers D under four simple models
Interpretation of Somers D under four simple models Roger B. Newson 03 September, 04 Introduction Somers D is an ordinal measure of association introduced by Somers (96)[9]. It can be defined in terms
More informationStatistical Machine Learning
Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes
More information4. Joint Distributions of Two Random Variables
4. Joint Distributions of Two Random Variables 4.1 Joint Distributions of Two Discrete Random Variables Suppose the discrete random variables X and Y have supports S X and S Y, respectively. The joint
More informationImperfect Debugging in Software Reliability
Imperfect Debugging in Software Reliability Tevfik Aktekin and Toros Caglar University of New Hampshire Peter T. Paul College of Business and Economics Department of Decision Sciences and United Health
More informationCHAPTER 3 EXAMPLES: REGRESSION AND PATH ANALYSIS
Examples: Regression And Path Analysis CHAPTER 3 EXAMPLES: REGRESSION AND PATH ANALYSIS Regression analysis with univariate or multivariate dependent variables is a standard procedure for modeling relationships
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
More informationProbability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0
Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0 This primer provides an overview of basic concepts and definitions in probability and statistics. We shall
More informationHypothesis Testing. 1 Introduction. 2 Hypotheses. 2.1 Null and Alternative Hypotheses. 2.2 Simple vs. Composite. 2.3 OneSided and TwoSided Tests
Hypothesis Testing 1 Introduction This document is a simple tutorial on hypothesis testing. It presents the basic concepts and definitions as well as some frequently asked questions associated with hypothesis
More informationAn Introduction to Using WinBUGS for CostEffectiveness Analyses in Health Economics
Slide 1 An Introduction to Using WinBUGS for CostEffectiveness Analyses in Health Economics Dr. Christian Asseburg Centre for Health Economics Part 1 Slide 2 Talk overview Foundations of Bayesian statistics
More informationAn Introduction to Machine Learning
An Introduction to Machine Learning L5: Novelty Detection and Regression Alexander J. Smola Statistical Machine Learning Program Canberra, ACT 0200 Australia Alex.Smola@nicta.com.au Tata Institute, Pune,
More informationTutorial on Markov Chain Monte Carlo
Tutorial on Markov Chain Monte Carlo Kenneth M. Hanson Los Alamos National Laboratory Presented at the 29 th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Technology,
More informationHandling attrition and nonresponse in longitudinal data
Longitudinal and Life Course Studies 2009 Volume 1 Issue 1 Pp 6372 Handling attrition and nonresponse in longitudinal data Harvey Goldstein University of Bristol Correspondence. Professor H. Goldstein
More informationMicroeconometrics Blundell Lecture 1 Overview and Binary Response Models
Microeconometrics Blundell Lecture 1 Overview and Binary Response Models Richard Blundell http://www.ucl.ac.uk/~uctp39a/ University College London FebruaryMarch 2016 Blundell (University College London)
More informationproblem arises when only a nonrandom sample is available differs from censored regression model in that x i is also unobserved
4 Data Issues 4.1 Truncated Regression population model y i = x i β + ε i, ε i N(0, σ 2 ) given a random sample, {y i, x i } N i=1, then OLS is consistent and efficient problem arises when only a nonrandom
More informationBasic Bayesian Methods
6 Basic Bayesian Methods Mark E. Glickman and David A. van Dyk Summary In this chapter, we introduce the basics of Bayesian data analysis. The key ingredients to a Bayesian analysis are the likelihood
More informationChapter 13 Introduction to Nonlinear Regression( 非 線 性 迴 歸 )
Chapter 13 Introduction to Nonlinear Regression( 非 線 性 迴 歸 ) and Neural Networks( 類 神 經 網 路 ) 許 湘 伶 Applied Linear Regression Models (Kutner, Nachtsheim, Neter, Li) hsuhl (NUK) LR Chap 10 1 / 35 13 Examples
More informationProbability Theory. Elementary rules of probability Sum rule. Product rule. p. 23
Probability Theory Uncertainty is key concept in machine learning. Probability provides consistent framework for the quantification and manipulation of uncertainty. Probability of an event is the fraction
More informationNumerical Methods for Option Pricing
Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly
More informationHURDLE AND SELECTION MODELS Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July 2009
HURDLE AND SELECTION MODELS Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July 2009 1. Introduction 2. A General Formulation 3. Truncated Normal Hurdle Model 4. Lognormal
More informationMaximum Likelihood Estimation
Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for
More informationSummary of Probability
Summary of Probability Mathematical Physics I Rules of Probability The probability of an event is called P(A), which is a positive number less than or equal to 1. The total probability for all possible
More informationA crash course in probability and Naïve Bayes classification
Probability theory A crash course in probability and Naïve Bayes classification Chapter 9 Random variable: a variable whose possible values are numerical outcomes of a random phenomenon. s: A person s
More informationAuxiliary Variables in Mixture Modeling: 3Step Approaches Using Mplus
Auxiliary Variables in Mixture Modeling: 3Step Approaches Using Mplus Tihomir Asparouhov and Bengt Muthén Mplus Web Notes: No. 15 Version 8, August 5, 2014 1 Abstract This paper discusses alternatives
More informationDURATION ANALYSIS OF FLEET DYNAMICS
DURATION ANALYSIS OF FLEET DYNAMICS Garth Holloway, University of Reading, garth.holloway@reading.ac.uk David Tomberlin, NOAA Fisheries, david.tomberlin@noaa.gov ABSTRACT Though long a standard technique
More informationOn Marginal Effects in Semiparametric Censored Regression Models
On Marginal Effects in Semiparametric Censored Regression Models Bo E. Honoré September 3, 2008 Introduction It is often argued that estimation of semiparametric censored regression models such as the
More informationAnalysis of Bayesian Dynamic Linear Models
Analysis of Bayesian Dynamic Linear Models Emily M. Casleton December 17, 2010 1 Introduction The main purpose of this project is to explore the Bayesian analysis of Dynamic Linear Models (DLMs). The main
More informationProbability and Statistics
CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b  0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute  Systems and Modeling GIGA  Bioinformatics ULg kristel.vansteen@ulg.ac.be
More informationDEPARTMENT OF ECONOMICS. Unit ECON 12122 Introduction to Econometrics. Notes 4 2. R and F tests
DEPARTMENT OF ECONOMICS Unit ECON 11 Introduction to Econometrics Notes 4 R and F tests These notes provide a summary of the lectures. They are not a complete account of the unit material. You should also
More informationSTAT 830 Convergence in Distribution
STAT 830 Convergence in Distribution Richard Lockhart Simon Fraser University STAT 830 Fall 2011 Richard Lockhart (Simon Fraser University) STAT 830 Convergence in Distribution STAT 830 Fall 2011 1 / 31
More informationMANBITESDOG BUSINESS CYCLES ONLINE APPENDIX
MANBITESDOG BUSINESS CYCLES ONLINE APPENDIX KRISTOFFER P. NIMARK The next section derives the equilibrium expressions for the beauty contest model from Section 3 of the main paper. This is followed by
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationIntroduction to latent variable models
Introduction to latent variable models lecture 1 Francesco Bartolucci Department of Economics, Finance and Statistics University of Perugia, IT bart@stat.unipg.it Outline [2/24] Latent variables and their
More informationIntroduction to General and Generalized Linear Models
Introduction to General and Generalized Linear Models General Linear Models  part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK2800 Kgs. Lyngby
More informationProbability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special DistributionsVI Today, I am going to introduce
More informationParameter estimation for nonlinear models: Numerical approaches to solving the inverse problem. Lecture 12 04/08/2008. Sven Zenker
Parameter estimation for nonlinear models: Numerical approaches to solving the inverse problem Lecture 12 04/08/2008 Sven Zenker Assignment no. 8 Correct setup of likelihood function One fixed set of observation
More informationNonparametric adaptive age replacement with a onecycle criterion
Nonparametric adaptive age replacement with a onecycle criterion P. CoolenSchrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK email: Pauline.Schrijner@durham.ac.uk
More informationA HYBRID GENETIC ALGORITHM FOR THE MAXIMUM LIKELIHOOD ESTIMATION OF MODELS WITH MULTIPLE EQUILIBRIA: A FIRST REPORT
New Mathematics and Natural Computation Vol. 1, No. 2 (2005) 295 303 c World Scientific Publishing Company A HYBRID GENETIC ALGORITHM FOR THE MAXIMUM LIKELIHOOD ESTIMATION OF MODELS WITH MULTIPLE EQUILIBRIA:
More informationP (x) 0. Discrete random variables Expected value. The expected value, mean or average of a random variable x is: xp (x) = v i P (v i )
Discrete random variables Probability mass function Given a discrete random variable X taking values in X = {v 1,..., v m }, its probability mass function P : X [0, 1] is defined as: P (v i ) = Pr[X =
More informationDepartment of Mathematics, Indian Institute of Technology, Kharagpur Assignment 23, Probability and Statistics, March 2015. Due:March 25, 2015.
Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment 3, Probability and Statistics, March 05. Due:March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x
More informationDetermining distribution parameters from quantiles
Determining distribution parameters from quantiles John D. Cook Department of Biostatistics The University of Texas M. D. Anderson Cancer Center P. O. Box 301402 Unit 1409 Houston, TX 772301402 USA cook@mderson.org
More informationModeling the Distribution of Environmental Radon Levels in Iowa: Combining Multiple Sources of Spatially Misaligned Data
Modeling the Distribution of Environmental Radon Levels in Iowa: Combining Multiple Sources of Spatially Misaligned Data Brian J. Smith, Ph.D. The University of Iowa Joint Statistical Meetings August 10,
More informationSupplement to Call Centers with Delay Information: Models and Insights
Supplement to Call Centers with Delay Information: Models and Insights Oualid Jouini 1 Zeynep Akşin 2 Yves Dallery 1 1 Laboratoire Genie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92290
More informationModelbased Synthesis. Tony O Hagan
Modelbased Synthesis Tony O Hagan Stochastic models Synthesising evidence through a statistical model 2 Evidence Synthesis (Session 3), Helsinki, 28/10/11 Graphical modelling The kinds of models that
More informationMarkov Chain Monte Carlo and Applied Bayesian Statistics: a short course Chris Holmes Professor of Biostatistics Oxford Centre for Gene Function
MCMC Appl. Bayes 1 Markov Chain Monte Carlo and Applied Bayesian Statistics: a short course Chris Holmes Professor of Biostatistics Oxford Centre for Gene Function MCMC Appl. Bayes 2 Objectives of Course
More information1 Gaussian Elimination
Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 GaussJordan reduction and the Reduced
More informationIntroduction to Markov Chain Monte Carlo
Introduction to Markov Chain Monte Carlo Monte Carlo: sample from a distribution to estimate the distribution to compute max, mean Markov Chain Monte Carlo: sampling using local information Generic problem
More informationPrinciple of Data Reduction
Chapter 6 Principle of Data Reduction 6.1 Introduction An experimenter uses the information in a sample X 1,..., X n to make inferences about an unknown parameter θ. If the sample size n is large, then
More information4. Joint Distributions
Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose
More informationCombining information from different survey samples  a case study with data collected by world wide web and telephone
Combining information from different survey samples  a case study with data collected by world wide web and telephone Magne Aldrin Norwegian Computing Center P.O. Box 114 Blindern N0314 Oslo Norway Email:
More informationEstimating Industry Multiples
Estimating Industry Multiples Malcolm Baker * Harvard University Richard S. Ruback Harvard University First Draft: May 1999 Rev. June 11, 1999 Abstract We analyze industry multiples for the S&P 500 in
More information3. Regression & Exponential Smoothing
3. Regression & Exponential Smoothing 3.1 Forecasting a Single Time Series Two main approaches are traditionally used to model a single time series z 1, z 2,..., z n 1. Models the observation z t as a
More informationUsing the Delta Method to Construct Confidence Intervals for Predicted Probabilities, Rates, and Discrete Changes
Using the Delta Method to Construct Confidence Intervals for Predicted Probabilities, Rates, Discrete Changes JunXuJ.ScottLong Indiana University August 22, 2005 The paper provides technical details on
More informationGamma Distribution Fitting
Chapter 552 Gamma Distribution Fitting Introduction This module fits the gamma probability distributions to a complete or censored set of individual or grouped data values. It outputs various statistics
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical
More informationMath 2020 Quizzes Winter 2009
Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile
More informationExample: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not.
Statistical Learning: Chapter 4 Classification 4.1 Introduction Supervised learning with a categorical (Qualitative) response Notation:  Feature vector X,  qualitative response Y, taking values in C
More information