Bayesian Modeling, Inference and Prediction
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Generalized Linear Models (Glms)
San Jos´eState University Math 261A: Regression Theory & Methods Generalized Linear Models (GLMs) Dr. Guangliang Chen This lecture is based on the following textbook sections: • Chapter 13: 13.1 – 13.3 Outline of this presentation: • What is a GLM? • Logistic regression • Poisson regression Generalized Linear Models (GLMs) What is a GLM? In ordinary linear regression, we assume that the response is a linear function of the regressors plus Gaussian noise: 0 2 y = β0 + β1x1 + ··· + βkxk + ∼ N(x β, σ ) | {z } |{z} linear form x0β N(0,σ2) noise The model can be reformulate in terms of • distribution of the response: y | x ∼ N(µ, σ2), and • dependence of the mean on the predictors: µ = E(y | x) = x0β Dr. Guangliang Chen | Mathematics & Statistics, San Jos´e State University3/24 Generalized Linear Models (GLMs) beta=(1,2) 5 4 3 β0 + β1x b y 2 y 1 0 −1 0.0 0.2 0.4 0.6 0.8 1.0 x x Dr. Guangliang Chen | Mathematics & Statistics, San Jos´e State University4/24 Generalized Linear Models (GLMs) Generalized linear models (GLM) extend linear regression by allowing the response variable to have • a general distribution (with mean µ = E(y | x)) and • a mean that depends on the predictors through a link function g: That is, g(µ) = β0x or equivalently, µ = g−1(β0x) Dr. Guangliang Chen | Mathematics & Statistics, San Jos´e State University5/24 Generalized Linear Models (GLMs) In GLM, the response is typically assumed to have a distribution in the exponential family, which is a large class of probability distributions that have pdfs of the form f(x | θ) = a(x)b(θ) exp(c(θ) · T (x)), including • Normal - ordinary linear regression • Bernoulli - Logistic regression, modeling binary data • Binomial - Multinomial logistic regression, modeling general cate- gorical data • Poisson - Poisson regression, modeling count data • Exponential, Gamma - survival analysis Dr. -
How Prediction Statistics Can Help Us Cope When We Are Shaken, Scared and Irrational
EGU21-15219 https://doi.org/10.5194/egusphere-egu21-15219 EGU General Assembly 2021 © Author(s) 2021. This work is distributed under the Creative Commons Attribution 4.0 License. How prediction statistics can help us cope when we are shaken, scared and irrational Yavor Kamer1, Shyam Nandan2, Stefan Hiemer1, Guy Ouillon3, and Didier Sornette4,5 1RichterX.com, Zürich, Switzerland 2Windeggstrasse 5, 8953 Dietikon, Zurich, Switzerland 3Lithophyse, Nice, France 4Department of Management,Technology and Economics, ETH Zürich, Zürich, Switzerland 5Institute of Risk Analysis, Prediction and Management (Risks-X), SUSTech, Shenzhen, China Nature is scary. You can be sitting at your home and next thing you know you are trapped under the ruble of your own house or sucked into a sinkhole. For millions of years we have been the figurines of this precarious scene and we have found our own ways of dealing with the anxiety. It is natural that we create and consume prophecies, conspiracies and false predictions. Information technologies amplify not only our rational but also irrational deeds. Social media algorithms, tuned to maximize attention, make sure that misinformation spreads much faster than its counterpart. What can we do to minimize the adverse effects of misinformation, especially in the case of earthquakes? One option could be to designate one authoritative institute, set up a big surveillance network and cancel or ban every source of misinformation before it spreads. This might have worked a few centuries ago but not in this day and age. Instead we propose a more inclusive option: embrace all voices and channel them into an actual, prospective earthquake prediction platform (Kamer et al. -
Reliability Engineering: Trends, Strategies and Best Practices
Reliability Engineering: Trends, Strategies and Best Practices WHITE PAPER September 2007 Predictive HCL’s Predictive Engineering encompasses the complete product life-cycle process, from concept to design to prototyping/testing, all the way to manufacturing. This helps in making decisions early Engineering in the design process, perfecting the product – thereby cutting down cycle time and costs, and Think. Design. Perfect! meeting set reliability and quality standards. Reliability Engineering: Trends, Strategies and Best Practices | September 2007 TABLE OF CONTENTS Abstract 3 Importance of reliability engineering in product industry 3 Current trends in reliability engineering 4 Reliability planning – an important task 5 Strength of reliability analysis 6 Is your reliability test plan optimized? 6 Challenges to overcome 7 Outsourcing of a reliability task 7 About HCL 10 © 2007, HCL Technologies. Reproduction Prohibited. This document is protected under Copyright by the Author, all rights reserved. Reliability Engineering: Trends, Strategies and Best Practices | September 2007 Abstract In reliability engineering, product industries now follow a conscious and planned approach to effectively address multiple issues related to product certification, failure returns, customer satisfaction, market competition and product lifecycle cost. Today, reliability professionals face new challenges posed by advanced and complex technology. Expertise and experience are necessary to provide an optimized solution meeting time and cost constraints associated with analysis and testing. In the changed scenario, the reliability approach has also become more objective and result oriented. This is well supported by analysis software. This paper discusses all associated issues including outsourcing of reliability tasks to a professional service provider as an alternate cost-effective option. Importance of reliability engineering in product industry The degree of importance given to the reliability of products varies depending on their criticality. -
R G M B.Ec. M.Sc. Ph.D
Rolando Kristiansand, Norway www.bayesgroup.org Gonzales Martinez +47 41224721 [email protected] JOBS & AWARDS M.Sc. B.Ec. Applied Ph.D. Economics candidate Statistics Jobs Events Awards Del Valle University University of Alcalá Universitetet i Agder 2018 Bolivia, 2000-2004 Spain, 2009-2010 Norway, 2018- 1/2018 Ph.D scholarship 8/2017 International Consultant, OPHI OTHER EDUCATION First prize, International Young 12/2016 J-PAL Course on Evaluating Social Programs Macro-prudential Policies Statistician Prize, IAOS Massachusetts Institute of Technology - MIT International Monetary Fund (IMF) 5/2016 Deputy manager, BCP bank Systematic Literature Reviews and Meta-analysis Theorizing and Theory Building Campbell foundation - Vrije Universiteit Halmstad University (HH, Sweden) 1/2016 Consultant, UNFPA Bayesian Modeling, Inference and Prediction Multidimensional Poverty Analysis University of Reading OPHI, Oxford University Researcher, data analyst & SKILLS AND TECHNOLOGIES 1/2015 project coordinator, PEP Academic Research & publishing Financial & Risk analysis 12 years of experience making economic/nancial analysis 3/2014 Director, BayesGroup.org and building mathemati- Consulting cal/statistical/econometric models for the government, private organizations and non- prot NGOs Research & project Mathematical 1/2013 modeling coordinator, UNFPA LANGUAGES 04/2012 First award, BCG competition Spanish (native) Statistical analysis English (TOEFL: 106) 9/2011 Financial Economist, UDAPE MatLab Macro-economic R, RStudio 11/2010 Currently, I work mainly analyst, MEFP Stata, SPSS with MatLab and R . SQL, SAS When needed, I use 3/2010 First award, BCB competition Eviews Stata, SAS or Eviews. I started working in LaTex, WinBugs Python recently. 8/2009 M.Sc. scholarship Python, Spyder RECENT PUBLICATIONS Disjunction between universality and targeting of social policies: Demographic opportunities for depolarized de- velopment in Paraguay. -
End-User Probabilistic Programming
End-User Probabilistic Programming Judith Borghouts, Andrew D. Gordon, Advait Sarkar, and Neil Toronto Microsoft Research Abstract. Probabilistic programming aims to help users make deci- sions under uncertainty. The user writes code representing a probabilistic model, and receives outcomes as distributions or summary statistics. We consider probabilistic programming for end-users, in particular spread- sheet users, estimated to number in tens to hundreds of millions. We examine the sources of uncertainty actually encountered by spreadsheet users, and their coping mechanisms, via an interview study. We examine spreadsheet-based interfaces and technology to help reason under uncer- tainty, via probabilistic and other means. We show how uncertain values can propagate uncertainty through spreadsheets, and how sheet-defined functions can be applied to handle uncertainty. Hence, we draw conclu- sions about the promise and limitations of probabilistic programming for end-users. 1 Introduction In this paper, we discuss the potential of bringing together two rather distinct approaches to decision making under uncertainty: spreadsheets and probabilistic programming. We start by introducing these two approaches. 1.1 Background: Spreadsheets and End-User Programming The spreadsheet is the first \killer app" of the personal computer era, starting in 1979 with Dan Bricklin and Bob Frankston's VisiCalc for the Apple II [15]. The primary interface of a spreadsheet|then and now, four decades later|is the grid, a two-dimensional array of cells. Each cell may hold a literal data value, or a formula that computes a data value (and may depend on the data in other cells, which may themselves be computed by formulas). -
Bayesian Data Analysis and Probabilistic Programming
Probabilistic programming and Stan mc-stan.org Outline What is probabilistic programming Stan now Stan in the future A bit about other software The next wave of probabilistic programming Stan even wider adoption including wider use in companies simulators (likelihood free) autonomus agents (reinforcmenet learning) deep learning Probabilistic programming Probabilistic programming framework BUGS started revolution in statistical analysis in early 1990’s allowed easier use of elaborate models by domain experts was widely adopted in many fields of science Probabilistic programming Probabilistic programming framework BUGS started revolution in statistical analysis in early 1990’s allowed easier use of elaborate models by domain experts was widely adopted in many fields of science The next wave of probabilistic programming Stan even wider adoption including wider use in companies simulators (likelihood free) autonomus agents (reinforcmenet learning) deep learning Stan Tens of thousands of users, 100+ contributors, 50+ R packages building on Stan Commercial and scientific users in e.g. ecology, pharmacometrics, physics, political science, finance and econometrics, professional sports, real estate Used in scientific breakthroughs: LIGO gravitational wave discovery (2017 Nobel Prize in Physics) and counting black holes Used in hundreds of companies: Facebook, Amazon, Google, Novartis, Astra Zeneca, Zalando, Smartly.io, Reaktor, ... StanCon 2018 organized in Helsinki in 29-31 August http://mc-stan.org/events/stancon2018Helsinki/ Probabilistic program -
Prediction in Multilevel Generalized Linear Models
J. R. Statist. Soc. A (2009) 172, Part 3, pp. 659–687 Prediction in multilevel generalized linear models Anders Skrondal Norwegian Institute of Public Health, Oslo, Norway and Sophia Rabe-Hesketh University of California, Berkeley, USA, and Institute of Education, London, UK [Received February 2008. Final revision October 2008] Summary. We discuss prediction of random effects and of expected responses in multilevel generalized linear models. Prediction of random effects is useful for instance in small area estimation and disease mapping, effectiveness studies and model diagnostics. Prediction of expected responses is useful for planning, model interpretation and diagnostics. For prediction of random effects, we concentrate on empirical Bayes prediction and discuss three different kinds of standard errors; the posterior standard deviation and the marginal prediction error standard deviation (comparative standard errors) and the marginal sampling standard devi- ation (diagnostic standard error). Analytical expressions are available only for linear models and are provided in an appendix. For other multilevel generalized linear models we present approximations and suggest using parametric bootstrapping to obtain standard errors. We also discuss prediction of expectations of responses or probabilities for a new unit in a hypotheti- cal cluster, or in a new (randomly sampled) cluster or in an existing cluster. The methods are implemented in gllamm and illustrated by applying them to survey data on reading proficiency of children nested in schools. Simulations are used to assess the performance of various pre- dictions and associated standard errors for logistic random-intercept models under a range of conditions. Keywords: Adaptive quadrature; Best linear unbiased predictor (BLUP); Comparative standard error; Diagnostic standard error; Empirical Bayes; Generalized linear mixed model; gllamm; Mean-squared error of prediction; Multilevel model; Posterior; Prediction; Random effects; Scoring 1. -
A Philosophical Treatise of Universal Induction
Entropy 2011, 13, 1076-1136; doi:10.3390/e13061076 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Article A Philosophical Treatise of Universal Induction Samuel Rathmanner and Marcus Hutter ? Research School of Computer Science, Australian National University, Corner of North and Daley Road, Canberra ACT 0200, Australia ? Author to whom correspondence should be addressed; E-Mail: [email protected]. Received: 20 April 2011; in revised form: 24 May 2011 / Accepted: 27 May 2011 / Published: 3 June 2011 Abstract: Understanding inductive reasoning is a problem that has engaged mankind for thousands of years. This problem is relevant to a wide range of fields and is integral to the philosophy of science. It has been tackled by many great minds ranging from philosophers to scientists to mathematicians, and more recently computer scientists. In this article we argue the case for Solomonoff Induction, a formal inductive framework which combines algorithmic information theory with the Bayesian framework. Although it achieves excellent theoretical results and is based on solid philosophical foundations, the requisite technical knowledge necessary for understanding this framework has caused it to remain largely unknown and unappreciated in the wider scientific community. The main contribution of this article is to convey Solomonoff induction and its related concepts in a generally accessible form with the aim of bridging this current technical gap. In the process we examine the major historical contributions that have led to the formulation of Solomonoff Induction as well as criticisms of Solomonoff and induction in general. In particular we examine how Solomonoff induction addresses many issues that have plagued other inductive systems, such as the black ravens paradox and the confirmation problem, and compare this approach with other recent approaches. -
Software Packages – Overview There Are Many Software Packages Available for Performing Statistical Analyses. the Ones Highlig
APPENDIX C SOFTWARE OPTIONS Software Packages – Overview There are many software packages available for performing statistical analyses. The ones highlighted here are selected because they provide comprehensive statistical design and analysis tools and a user friendly interface. SAS SAS is arguably one of the most comprehensive packages in terms of statistical analyses. However, the individual license cost is expensive. Primarily, SAS is a macro based scripting language, making it challenging for new users. However, it provides a comprehensive analysis ranging from the simple t-test to Generalized Linear mixed models. SAS also provides limited Design of Experiments capability. SAS supports factorial designs and optimal designs. However, it does not handle design constraints well. SAS also has a GUI available: SAS Enterprise, however, it has limited capability. SAS is a good program to purchase if your organization conducts advanced statistical analyses or works with very large datasets. R R, similar to SAS, provides a comprehensive analysis toolbox. R is an open source software package that has been approved for use on some DoD computer (contact AFFTC for details). Because R is open source its price is very attractive (free). However, the help guides are not user friendly. R is primarily a scripting software with notation similar to MatLab. R’s GUI: R Commander provides limited DOE capabilities. R is a good language to download if you can get it approved for use and you have analysts with a strong programming background. MatLab w/ Statistical Toolbox MatLab has a statistical toolbox that provides a wide variety of statistical analyses. The analyses range from simple hypotheses test to complex models and the ability to use likelihood maximization. -
Richard C. Zink, Ph.D
Richard C. Zink, Ph.D. SAS Institute, Inc. • SAS Campus Drive • Cary, North Carolina 27513 • Office 919.531.4710 • Mobile 919.397.4238 • Blog: http://blogs.sas.com/content/jmp/author/rizink/ • [email protected] • @rczink SUMMARY Richard C. Zink is Principal Research Statistician Developer in the JMP Life Sciences division at SAS Institute. He is currently a developer for JMP Clinical, an innovative software package designed to streamline the review of clinical trial data. Richard joined SAS in 2011 after eight years in the pharmaceutical industry, where he designed and analyzed clinical trials in diverse therapeutic areas including infectious disease, oncology, and ophthalmology, and participated in US and European drug submissions and FDA advisory committee hearings. Richard is the Publications Officer for the Biopharmaceutical Section of the American Statistical Association, and the Statistics Section Editor for the Therapeutic Innovation & Regulatory Science (formerly Drug Information Journal). He holds a Ph.D. in Biostatistics from the University of North Carolina at Chapel Hill, where he serves as an Adjunct Assistant Professor of Biostatistics. His research interests include data visualization, the analysis of pre- and post-market adverse events, subgroup identification for patients with enhanced treatment response, and the assessment of data integrity in clinical trials. He is author of Risk-Based Monitoring and Fraud Detection in Clinical Trials Using JMP and SAS, and co-editor of Modern Approaches to Clinical Trials Using -
Winbugs for Beginners
WinBUGS for Beginners Gabriela Espino-Hernandez Department of Statistics UBC July 2010 “A knowledge of Bayesian statistics is assumed…” The content of this presentation is mainly based on WinBUGS manual Introduction BUGS 1: “Bayesian inference Using Gibbs Sampling” Project for Bayesian analysis using MCMC methods It is not being further developed WinBUGS 1,2 Stable version Run directly from R and other programs OpenBUGS 3 Currently experimental Run directly from R and other programs Running under Linux as LinBUGS 1 MRC Biostatistics Unit Cambridge, 2 Imperial College School of Medicine at St Mary's, London 3 University of Helsinki, Finland WinBUGS Freely distributed http://www.mrc-bsu.cam.ac.uk/bugs/welcome.shtml Key for unrestricted use http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/WinBUGS14_immortality_key.txt WinBUGS installation also contains: Extensive user manual Examples Control analysis using: Standard windows interface DoodleBUGS: Graphical representation of model A closed form for the posterior distribution is not needed Conditional independence is assumed Improper priors are not allowed Inputs Model code Specify data distributions Specify parameter distributions (priors) Data List / rectangular format Initial values for parameters Load / generate Model specification model { statements to describe model in BUGS language } Multiple statements in a single line or one statement over several lines Comment line is followed by # Types of nodes 1. Stochastic Variables that are given a distribution 2. Deterministic -
Some Statistical Models for Prediction Jonathan Auerbach Submitted In
Some Statistical Models for Prediction Jonathan Auerbach Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy under the Executive Committee of the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2020 © 2020 Jonathan Auerbach All Rights Reserved Abstract Some Statistical Models for Prediction Jonathan Auerbach This dissertation examines the use of statistical models for prediction. Examples are drawn from public policy and chosen because they represent pressing problems facing U.S. governments at the local, state, and federal level. The first five chapters provide examples where the perfunctory use of linear models, the prediction tool of choice in government, failed to produce reasonable predictions. Methodological flaws are identified, and more accurate models are proposed that draw on advances in statistics, data science, and machine learning. Chapter 1 examines skyscraper construction, where the normality assumption is violated and extreme value analysis is more appropriate. Chapters 2 and 3 examine presidential approval and voting (a leading measure of civic participation), where the non-collinearity assumption is violated and an index model is more appropriate. Chapter 4 examines changes in temperature sensitivity due to global warming, where the linearity assumption is violated and a first-hitting-time model is more appropriate. Chapter 5 examines the crime rate, where the independence assumption is violated and a block model is more appropriate. The last chapter provides an example where simple linear regression was overlooked as providing a sensible solution. Chapter 6 examines traffic fatalities, where the linear assumption provides a better predictor than the more popular non-linear probability model, logistic regression.