BUGS Code for Item Response Theory
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Chapter 7 Assessing and Improving Convergence of the Markov Chain
Chapter 7 Assessing and improving convergence of the Markov chain Questions: . Are we getting the right answer? . Can we get the answer quicker? Bayesian Biostatistics - Piracicaba 2014 376 7.1 Introduction • MCMC sampling is powerful, but comes with a cost: dependent sampling + checking convergence is not easy: ◦ Convergence theorems do not tell us when convergence will occur ◦ In this chapter: graphical + formal diagnostics to assess convergence • Acceleration techniques to speed up MCMC sampling procedure • Data augmentation as Bayesian generalization of EM algorithm Bayesian Biostatistics - Piracicaba 2014 377 7.2 Assessing convergence of a Markov chain Bayesian Biostatistics - Piracicaba 2014 378 7.2.1 Definition of convergence for a Markov chain Loose definition: With increasing number of iterations (k −! 1), the distribution of k θ , pk(θ), converges to the target distribution p(θ j y). In practice, convergence means: • Histogram of θk remains the same along the chain • Summary statistics of θk remain the same along the chain Bayesian Biostatistics - Piracicaba 2014 379 Approaches • Theoretical research ◦ Establish conditions that ensure convergence, but in general these theoretical results cannot be used in practice • Two types of practical procedures to check convergence: ◦ Checking stationarity: from which iteration (k0) is the chain sampling from the posterior distribution (assessing burn-in part of the Markov chain). ◦ Checking accuracy: verify that the posterior summary measures are computed with the desired accuracy • Most -
Getting Started in Openbugs / Winbugs
Practical 1: Getting started in OpenBUGS Slide 1 An Introduction to Using WinBUGS for Cost-Effectiveness Analyses in Health Economics Dr. Christian Asseburg Centre for Health Economics University of York, UK [email protected] Practical 1 Getting started in OpenBUGS / WinB UGS 2007-03-12, Linköping Dr Christian Asseburg University of York, UK [email protected] Practical 1: Getting started in OpenBUGS Slide 2 ● Brief comparison WinBUGS / OpenBUGS ● Practical – Opening OpenBUGS – Entering a model and data – Some error messages – Starting the sampler – Checking sampling performance – Retrieving the posterior summaries 2007-03-12, Linköping Dr Christian Asseburg University of York, UK [email protected] Practical 1: Getting started in OpenBUGS Slide 3 WinBUGS / OpenBUGS ● WinBUGS was developed at the MRC Biostatistics unit in Cambridge. Free download, but registration required for a licence. No fee and no warranty. ● OpenBUGS is the current development of WinBUGS after its source code was released to the public. Download is free, no registration, GNU GPL licence. 2007-03-12, Linköping Dr Christian Asseburg University of York, UK [email protected] Practical 1: Getting started in OpenBUGS Slide 4 WinBUGS / OpenBUGS ● There are no major differences between the latest WinBUGS (1.4.1) and OpenBUGS (2.2.0) releases. ● Minor differences include: – OpenBUGS is occasionally a bit slower – WinBUGS requires Microsoft Windows OS – OpenBUGS error messages are sometimes more informative ● The examples in these slides use OpenBUGS. 2007-03-12, Linköping Dr Christian Asseburg University of York, UK [email protected] Practical 1: Getting started in OpenBUGS Slide 5 Practical 1: Target ● Start OpenBUGS ● Code the example from health economics from the earlier presentation ● Run the sampler and obtain posterior summaries 2007-03-12, Linköping Dr Christian Asseburg University of York, UK [email protected] Practical 1: Getting started in OpenBUGS Slide 6 Starting OpenBUGS ● You can download OpenBUGS from http://mathstat.helsinki.fi/openbugs/ ● Start the program. -
Stan: a Probabilistic Programming Language
JSS Journal of Statistical Software MMMMMM YYYY, Volume VV, Issue II. http://www.jstatsoft.org/ Stan: A Probabilistic Programming Language Bob Carpenter Andrew Gelman Matt Hoffman Columbia University Columbia University Adobe Research Daniel Lee Ben Goodrich Michael Betancourt Columbia University Columbia University University of Warwick Marcus A. Brubaker Jiqiang Guo Peter Li University of Toronto, NPD Group Columbia University Scarborough Allen Riddell Dartmouth College Abstract Stan is a probabilistic programming language for specifying statistical models. A Stan program imperatively defines a log probability function over parameters conditioned on specified data and constants. As of version 2.2.0, Stan provides full Bayesian inference for continuous-variable models through Markov chain Monte Carlo methods such as the No-U-Turn sampler, an adaptive form of Hamiltonian Monte Carlo sampling. Penalized maximum likelihood estimates are calculated using optimization methods such as the Broyden-Fletcher-Goldfarb-Shanno algorithm. Stan is also a platform for computing log densities and their gradients and Hessians, which can be used in alternative algorithms such as variational Bayes, expectation propa- gation, and marginal inference using approximate integration. To this end, Stan is set up so that the densities, gradients, and Hessians, along with intermediate quantities of the algorithm such as acceptance probabilities, are easily accessible. Stan can be called from the command line, through R using the RStan package, or through Python using the PyStan package. All three interfaces support sampling and optimization-based inference. RStan and PyStan also provide access to log probabilities, gradients, Hessians, and data I/O. Keywords: probabilistic program, Bayesian inference, algorithmic differentiation, Stan. -
Installing BUGS and the R to BUGS Interface 1. Brief Overview
File = E:\bugs\installing.bugs.jags.docm 1 John Miyamoto (email: [email protected]) Installing BUGS and the R to BUGS Interface Caveat: I am a Windows user so these notes are focused on Windows 7 installations. I will include what I know about the Mac and Linux versions of these programs but I cannot swear to the accuracy of my comments. Contents (Cntrl-left click on a link to jump to the corresponding section) Section Topic 1 Brief Overview 2 Installing OpenBUGS 3 Installing WinBUGs (Windows only) 4 OpenBUGS versus WinBUGS 5 Installing JAGS 6 Installing R packages that are used with OpenBUGS, WinBUGS, and JAGS 7 Running BRugs on a 32-bit or 64-bit Windows computer 8 Hints for Mac and Linux Users 9 References # End of Contents Table 1. Brief Overview TOC BUGS stands for Bayesian Inference Under Gibbs Sampling1. The BUGS program serves two critical functions in Bayesian statistics. First, given appropriate inputs, it computes the posterior distribution over model parameters - this is critical in any Bayesian statistical analysis. Second, it allows the user to compute a Bayesian analysis without requiring extensive knowledge of the mathematical analysis and computer programming required for the analysis. The user does need to understand the statistical model or models that are being analyzed, the assumptions that are made about model parameters including their prior distributions, and the structure of the data, but the BUGS program relieves the user of the necessity of creating an algorithm to sample from the posterior distribution and the necessity of writing the computer program that computes this algorithm. -
Winbugs Lectures X4.Pdf
Introduction to Bayesian Analysis and WinBUGS Summary 1. Probability as a means of representing uncertainty 2. Bayesian direct probability statements about parameters Lecture 1. 3. Probability distributions Introduction to Bayesian Monte Carlo 4. Monte Carlo simulation methods in WINBUGS 5. Implementation in WinBUGS (and DoodleBUGS) - Demo 6. Directed graphs for representing probability models 7. Examples 1-1 1-2 Introduction to Bayesian Analysis and WinBUGS Introduction to Bayesian Analysis and WinBUGS How did it all start? Basic idea: Direct expression of uncertainty about In 1763, Reverend Thomas Bayes of Tunbridge Wells wrote unknown parameters eg ”There is an 89% probability that the absolute increase in major bleeds is less than 10 percent with low-dose PLT transfusions” (Tinmouth et al, Transfusion, 2004) !50 !40 !30 !20 !10 0 10 20 30 % absolute increase in major bleeds In modern language, given r Binomial(θ,n), what is Pr(θ1 < θ < θ2 r, n)? ∼ | 1-3 1-4 Introduction to Bayesian Analysis and WinBUGS Introduction to Bayesian Analysis and WinBUGS Why a direct probability distribution? Inference on proportions 1. Tells us what we want: what are plausible values for the parameter of interest? What is a reasonable form for a prior distribution for a proportion? θ Beta[a, b] represents a beta distribution with properties: 2. No P-values: just calculate relevant tail areas ∼ Γ(a + b) a 1 b 1 p(θ a, b)= θ − (1 θ) − ; θ (0, 1) 3. No (difficult to interpret) confidence intervals: just report, say, central area | Γ(a)Γ(b) − ∈ a that contains 95% of distribution E(θ a, b)= | a + b ab 4. -
A Quick Reference to C Programming Language
A Quick Reference to C Programming Language Structure of a C Program #include(stdio.h) /* include IO library */ #include... /* include other files */ #define.. /* define constants */ /* Declare global variables*/) (variable type)(variable list); /* Define program functions */ (type returned)(function name)(parameter list) (declaration of parameter types) { (declaration of local variables); (body of function code); } /* Define main function*/ main ((optional argc and argv arguments)) (optional declaration parameters) { (declaration of local variables); (body of main function code); } Comments Format: /*(body of comment) */ Example: /*This is a comment in C*/ Constant Declarations Format: #define(constant name)(constant value) Example: #define MAXIMUM 1000 Type Definitions Format: typedef(datatype)(symbolic name); Example: typedef int KILOGRAMS; Variables Declarations: Format: (variable type)(name 1)(name 2),...; Example: int firstnum, secondnum; char alpha; int firstarray[10]; int doublearray[2][5]; char firststring[1O]; Initializing: Format: (variable type)(name)=(value); Example: int firstnum=5; Assignments: Format: (name)=(value); Example: firstnum=5; Alpha='a'; Unions Declarations: Format: union(tag) {(type)(member name); (type)(member name); ... }(variable name); Example: union demotagname {int a; float b; }demovarname; Assignment: Format: (tag).(member name)=(value); demovarname.a=1; demovarname.b=4.6; Structures Declarations: Format: struct(tag) {(type)(variable); (type)(variable); ... }(variable list); Example: struct student {int -
Lecture 2: Introduction to C Programming Language [email protected]
CSCI-UA 201 Joanna Klukowska Lecture 2: Introduction to C Programming Language [email protected] Lecture 2: Introduction to C Programming Language Notes include some materials provided by Andrew Case, Jinyang Li, Mohamed Zahran, and the textbooks. Reading materials Chapters 1-6 in The C Programming Language, by B.W. Kernighan and Dennis M. Ritchie Section 1.2 and Aside on page 4 in Computer Systems, A Programmer’s Perspective by R.E. Bryant and D.R. O’Hallaron Contents 1 Intro to C and Unix/Linux 3 1.1 Why C?............................................................3 1.2 C vs. Java...........................................................3 1.3 Code Development Process (Not Only in C).........................................4 1.4 Basic Unix Commands....................................................4 2 First C Program and Stages of Compilation 6 2.1 Writing and running hello world in C.............................................6 2.2 Hello world line by line....................................................7 2.3 What really happens when we compile a program?.....................................8 3 Basics of C 9 3.1 Data types (primitive types)..................................................9 3.2 Using printf to print different data types.........................................9 3.3 Control flow.......................................................... 10 3.4 Functions........................................................... 11 3.5 Variable scope........................................................ 12 3.6 Header files......................................................... -
A Tutorial Introduction to the Language B
A TUTORIAL INTRODUCTION TO THE LANGUAGE B B. W. Kernighan Bell Laboratories Murray Hill, New Jersey 1. Introduction B is a new computer language designed and implemented at Murray Hill. It runs and is actively supported and documented on the H6070 TSS system at Murray Hill. B is particularly suited for non-numeric computations, typified by system programming. These usually involve many complex logical decisions, computations on integers and fields of words, especially charac- ters and bit strings, and no floating point. B programs for such operations are substantially easier to write and understand than GMAP programs. The generated code is quite good. Implementation of simple TSS subsystems is an especially good use for B. B is reminiscent of BCPL [2] , for those who can remember. The original design and implementation are the work of K. L. Thompson and D. M. Ritchie; their original 6070 version has been substantially improved by S. C. Johnson, who also wrote the runtime library. This memo is a tutorial to make learning B as painless as possible. Most of the features of the language are mentioned in passing, but only the most important are stressed. Users who would like the full story should consult A User’s Reference to B on MH-TSS, by S. C. Johnson [1], which should be read for details any- way. We will assume that the user is familiar with the mysteries of TSS, such as creating files, text editing, and the like, and has programmed in some language before. Throughout, the symbolism (->n) implies that a topic will be expanded upon in section n of this manual. -
Flexible Programming of Hierarchical Modeling Algorithms and Compilation of R Using NIMBLE Christopher Paciorek UC Berkeley Statistics
Flexible Programming of Hierarchical Modeling Algorithms and Compilation of R Using NIMBLE Christopher Paciorek UC Berkeley Statistics Joint work with: Perry de Valpine (PI) UC Berkeley Environmental Science, Policy and Managem’t Daniel Turek Williams College Math and Statistics Nick Michaud UC Berkeley ESPM and Statistics Duncan Temple Lang UC Davis Statistics Bayesian nonparametrics development with: Claudia Wehrhahn Cortes UC Santa Cruz Applied Math and Statistics Abel Rodriguez UC Santa Cruz Applied Math and Statistics http://r-nimble.org October 2017 Funded by NSF DBI-1147230, ACI-1550488, DMS-1622444; Google Summer of Code 2015, 2017 Hierarchical statistical models A basic random effects / Bayesian hierarchical model Probabilistic model Flexible programming of hierarchical modeling algorithms using NIMBLE (r- 2 nimble.org) Hierarchical statistical models A basic random effects / Bayesian hierarchical model BUGS DSL code Probabilistic model # priors on hyperparameters alpha ~ dexp(1.0) beta ~ dgamma(0.1,1.0) for (i in 1:N){ # latent process (random effects) # random effects distribution theta[i] ~ dgamma(alpha,beta) # linear predictor lambda[i] <- theta[i]*t[i] # likelihood (data model) x[i] ~ dpois(lambda[i]) } Flexible programming of hierarchical modeling algorithms using NIMBLE (r- 3 nimble.org) Divorcing model specification from algorithm MCMC Flavor 1 Your new method Y(1) Y(2) Y(3) MCMC Flavor 2 Variational Bayes X(1) X(2) X(3) Particle Filter MCEM Quadrature Importance Sampler Maximum likelihood Flexible programming of hierarchical modeling algorithms using NIMBLE (r- 4 nimble.org) What can a practitioner do with hierarchical models? Two basic software designs: 1. Typical R/Python package = Model family + 1 or more algorithms • GLMMs: lme4, MCMCglmm • GAMMs: mgcv • spatial models: spBayes, INLA Flexible programming of hierarchical modeling algorithms using NIMBLE (r- 5 nimble.org) What can a practitioner do with hierarchical models? Two basic software designs: 1. -
Command $Line; Done
http://xkcd.com/208/ >0 TGCAGGTATATCTATTAGCAGGTTTAATTTTGCCTGCACTTGGTTGGGTACATTATTTTAAGTGTATTTGACAAG >1 TGCAGGTTGTTGTTACTCAGGTCCAGTTCTCTGAGACTGGAGGACTGGGAGCTGAGAACTGAGGACAGAGCTTCA >2 TGCAGGGCCGGTCCAAGGCTGCATGAGGCCTGGGGCAGAATCTGACCTAGGGGCCCCTCTTGCTGCTAAAACCAT >3 TGCAGGATCTGCTGCACCATTAACCAGACAGAAATGGCAGTTTTATACAAGTTATTATTCTAATTCAATAGCTGA >4 TGCAGGGGTCAAATACAGCTGTCAAAGCCAGACTTTGAGCACTGCTAGCTGGCTGCAACACCTGCACTTAACCTC cat seqs.fa PIPE grep ACGT TGCAGGTATATCTATTAGCAGGTTTAATTTTGCCTGCACTTGGTTGGGTACATTATTTTAAGTGTATTTGACAAG >1 TGCAGGTTGTTGTTACTCAGGTCCAGTTCTCTGAGACTGGAGGACTGGGAGCTGAGAACTGAGGACAGAGCTTCA >2 TGCAGGGCCGGTCCAAGGCTGCATGAGGCCTGGGGCAGAATCTGACCTAGGGGCCCCTCTTGCTGCTAAAACCAT >3 TGCAGGATCTGCTGCACCATTAACCAGACAGAAATGGCAGTTTTATACAAGTTATTATTCTAATTCAATAGCTGA >4 TGCAGGGGTCAAATACAGCTGTCAAAGCCAGACTTTGAGCACTGCTAGCTGGCTGCAACACCTGCACTTAACCTC cat seqs.fa Does PIPE “>0” grep ACGT contain “ACGT”? Yes? No? Output NULL >1 TGCAGGTTGTTGTTACTCAGGTCCAGTTCTCTGAGACTGGAGGACTGGGAGCTGAGAACTGAGGACAGAGCTTCA >2 TGCAGGGCCGGTCCAAGGCTGCATGAGGCCTGGGGCAGAATCTGACCTAGGGGCCCCTCTTGCTGCTAAAACCAT >3 TGCAGGATCTGCTGCACCATTAACCAGACAGAAATGGCAGTTTTATACAAGTTATTATTCTAATTCAATAGCTGA >4 TGCAGGGGTCAAATACAGCTGTCAAAGCCAGACTTTGAGCACTGCTAGCTGGCTGCAACACCTGCACTTAACCTC cat seqs.fa Does PIPE “TGCAGGTATATCTATTAGCAGGTTTAATTTTGCCTGCACTTG...G” grep ACGT contain “ACGT”? Yes? No? Output NULL TGCAGGTTGTTGTTACTCAGGTCCAGTTCTCTGAGACTGGAGGACTGGGAGCTGAGAACTGAGGACAGAGCTTCA >2 TGCAGGGCCGGTCCAAGGCTGCATGAGGCCTGGGGCAGAATCTGACCTAGGGGCCCCTCTTGCTGCTAAAACCAT >3 TGCAGGATCTGCTGCACCATTAACCAGACAGAAATGGCAGTTTTATACAAGTTATTATTCTAATTCAATAGCTGA -
Spaio-‐Temporal Dependence: a Blessing and a Curse For
Spao-temporal dependence: a blessing and a curse for computaon and inference (illustrated by composi$onal data modeling) (and with an introduc$on to NIMBLE) Christopher Paciorek UC Berkeley Stas$cs Joint work with: The PalEON project team (hp://paleonproject.org) The NIMBLE development team (hp://r-nimble.org) NSF-CBMS Workshop on Bayesian Spaal Stas$cs August 2017 Funded by various NSF grants to the PalEON and NIMBLE projects PalEON Project Goal: Improve the predic$ve capacity of terrestrial ecosystem models Friedlingstein et al. 2006 J. Climate “This large varia-on among carbon-cycle models … has been called ‘uncertainty’. I prefer to call it ‘ignorance’.” - Pren-ce (2013) Grantham Ins-tute Cri-cal issue: model parameterizaon and representaon of decadal- to centennial-scale processes are poorly constrained by data Approach: use historical and fossil data to es$mate past vegetaon and climate and use this informaon for model ini$alizaon, assessment, and improvement Spao-temporal dependence: a blessing 2 and a curse for computaon and inference Fossil Pollen Data Berry Pond West Berry Pond 2000 Year AD , Pine W Massachusetts 1900 Hemlock 1800 Birch 1700 1600 Oak 1500 Hickory 1400 Beech 1300 Chestnut 1200 Grass & weeds 1100 Charcoal:Pollen 1000 % organic matter Spao-temporal dependence: a blessing European settlement and a curse for computaon and inference 20 20 40 20 20 40 20 1500 60 O n set of Little Ice Age 3 SeQlement-era Land Survey Data Survey grid in Wisconsin Surveyor notes Raw oak tree propor$ons (on a grid in the western por$on -
The C Programming Language
The C programming Language The C programming Language By Brian W. Kernighan and Dennis M. Ritchie. Published by Prentice-Hall in 1988 ISBN 0-13-110362-8 (paperback) ISBN 0-13-110370-9 Contents ● Preface ● Preface to the first edition ● Introduction 1. Chapter 1: A Tutorial Introduction 1. Getting Started 2. Variables and Arithmetic Expressions 3. The for statement 4. Symbolic Constants 5. Character Input and Output 1. File Copying 2. Character Counting 3. Line Counting 4. Word Counting 6. Arrays 7. Functions 8. Arguments - Call by Value 9. Character Arrays 10. External Variables and Scope 2. Chapter 2: Types, Operators and Expressions 1. Variable Names 2. Data Types and Sizes 3. Constants 4. Declarations http://freebooks.by.ru/view/CProgrammingLanguage/kandr.html (1 of 5) [5/15/2002 10:12:59 PM] The C programming Language 5. Arithmetic Operators 6. Relational and Logical Operators 7. Type Conversions 8. Increment and Decrement Operators 9. Bitwise Operators 10. Assignment Operators and Expressions 11. Conditional Expressions 12. Precedence and Order of Evaluation 3. Chapter 3: Control Flow 1. Statements and Blocks 2. If-Else 3. Else-If 4. Switch 5. Loops - While and For 6. Loops - Do-While 7. Break and Continue 8. Goto and labels 4. Chapter 4: Functions and Program Structure 1. Basics of Functions 2. Functions Returning Non-integers 3. External Variables 4. Scope Rules 5. Header Files 6. Static Variables 7. Register Variables 8. Block Structure 9. Initialization 10. Recursion 11. The C Preprocessor 1. File Inclusion 2. Macro Substitution 3. Conditional Inclusion 5. Chapter 5: Pointers and Arrays 1.