Bayesian Statistics, Modeling & Reasoning What is this course about? P548: Intro Bayesian Stats with Psych Applications Instructor: John Miyamoto 01/04/2016: Lecture 01-1 Note: This Powerpoint presentation may contain macros that I wrote to help me create the slides. The macros aren’t needed to view the slides. You can disable or delete the macros without any change to the presentation.

Outline What is Bayesian inference? Why are Bayesian statistics, modeling & reasoning relevant to psychology? What is Psych 548 about? Explain Psych 548 website Intro to R Intro to RStudio Intro to the R to BUGS interface Psych 548, Miyamoto, Win '16 2 Lecture probably ends here

Bayes Rule – What Is It? Reverend Thomas Bayes, 1702 – 1761 English Protestant minister & mathematician Bayes Rule is fundamentally important to: ♦ Bayesian statistics ♦ Bayesian decision theory ♦ Bayesian models in psychology Psych 548, Miyamoto, Win '16 3 Bayes Rule – Why Is It Important?

Psych 548, Miyamoto, Win '16 4 Bayes Rule – Why Is It Important? Bayes Rule is the optimal way to update the probability of hypotheses given data. The concept of "Bayesian reasoning“: 3 related concepts ♦ Concept 1: Bayesian inference is a model of optimal learning from experience. ♦ Concept 2: Bayesian decision theory describes optimal strategies for taking actions in an uncertain environment. Optimal gambling strategies. ♦ Concept 3: Bayesian reasoning represents the uncertainty of events as probabilities in a mathematical calculus. Concepts 1, 2 & 3 are all consistent with the use of the term, "Bayesian", in modern psychology. Bayesian Issues in Psychology

Psych 548, Miyamoto, Win '16 5 Bayesian Issues in Psychological Research Does human reasoning about uncertainty conform to Bayes Rule? Do humans reason about uncertainty as if they are manipulating probabilities? ♦ These questions are posed with respect to infants & children, as well as adults. Do neural information processing systems (NIPS) incorporate Bayes Rule? Do NIPS model uncertainties as if they are probabilities. Four Roles for Bayesian Reasoning in Psychology Research

Psych 548, Miyamoto, Win '16 6 Four Roles for Bayesian Reasoning in Psychology 1.Bayesian statistics: Analyzing data ♦ E.g., is the slope of the regression of grades on IQ the same for boys as for girls? ♦ E.g., are there group differences in an analysis of variance? Four Roles …. (Continued)

Psych 548, Miyamoto, Win '16 7 Four Roles for Bayesian Reasoning in Psychology 1.Bayesian statistics: Analyzing data 2.Bayesian decision theory – a theory of strategic action. How to gamble if you must. 3.Bayesian modeling of psychological processes 4.Bayesian reasoning – Do people reason as if they are Bayesian probability analysts? (At macro & neural levels) ♦ Judgment and decision making – This is a major issue. ♦ Human causal reasoning – is it Bayesian or quasi-Bayesian? ♦ Modeling neural decision making – many proposed models have a strong Bayesian flavor. Four Roles …. (Continued)

Psych 548, Miyamoto, Win '16 8 Four Roles for Bayesian Reasoning in Psychology 1.Bayesian statistics: Analyzing data 2.Bayesian decision theory – a theory of strategic action. How to gamble if you must. 3.Bayesian modeling of psychological processes 4.Bayesian reasoning – Do people reason as if they are Bayesian probability analysts? (At macro & neural levels) Graphical Representation of Psych 548 Focus on Stats/Modeling Psych 548: Focus on Topics (1) and (3). Include a little bit of (4).

Psych 548, Miyamoto, Win '16 9 Graphical Representation of Psych 548 Bayesian Statistics & Modeling: R & JAGS Bayesian Models in Cognitive Psychology & Neuroscience Psych 548 Graph & Text Showing the History of S, S-Plus & R

Psych 548, Miyamoto, Win '16 10 Brief History of S, S-Plus, & R S – open source statistics program created by Bell Labs (1976 – 1988 – 1999) S-Plus – commercial statistics program, refinement of S (1988 – present) R – free open source statistics program (1997 – present) ♦ currently the standard computing framework for statisticians worldwide Many contributors to its development ♦ Excellent general computation. Powerful & flexible. ♦ Great graphics. ♦ Multiplatform: Unix, Linux, Windows, Mac ♦ User must like programming BUGS, WinBUGS, OpenBUGS, JAGS S S-Plus R Ancestry of R

Psych 548, Miyamoto, Win '16 11 BUGS, WinBUGS, OpenBUGS & JAGS Gibbs Sampling & Metropolis-Hastings Algorithm Two algorithms for sampling from a hard-to-evaluate probability distribution. BUGS – Bayesian inference Under Gibbs Sampling (circa 1995) WinBUGS - Open source (circa 1997) ♦ Windows only OpenBUGS – Open source (circa 2006) ♦ Mainly Windows. Runs within a virtual Windows machine on a Mac. JAGS – Open source (circa 2007) ♦ Multiplatform: Windows, Mac, Linux STAN – Open source (circa 2012) ○ Multiplatform: Windows, Mac, Linux Basic Structure of Bayesian Computation with R & OpenBUGS “BUGS” includes all of these.

Psych 548, Miyamoto, Win '16 12 Basic Structure of Bayesian Computation R data preparation analysis of results JAGS Computes approximation to the posterior distribution. Includes diagnostics. rjags functions rjags runjags OpenBUGS/ WinBUGS/ Stan R BRugs functions Brugs functions BRugs R2WinBUGS rstan Outline of Remainder of the Lecture: Course Outline & General Information

RStudio Run RStudio Run R from within RStudio Psych 548, Miyamoto, Win '16 13

Psych 548, Miyamoto, Win '16 14 Remainder of This Lecture Take 5 minute break Introduce selves Psych 548: What will we study? Briefly view the Psych 548 webpage. Introduction to the computer facility in CSSCR. Introduction to R, BUGS (OpenBUGS & JAGS), and RStudio 5 Minute Break

Introduce selves upon return Psych 548, Miyamoto, Win '16 15 Course Goals

Psych 548, Miyamoto, Win '16 16 Course Goals Learn the theoretical framework of Bayesian inference. Achieve competence with R, OpenBUGS and JAGS. Learn basic Bayesian statistics ♦ Learn how to think about statistical inference from a Bayesian standpoint. ♦ Learn how to interpret the results of a Bayesian analysis. ♦ Learn basic tools of Bayesian statistical inference - testing for convergence, making standard plots, examing samples from a posterior distribution Secondary Goals ♦ Bayesian modeling in psychology ♦ Understand arguments about Bayesian reasoning in the psychology of reasoning. The pros and cons of the heuristics & biases movement. Kruschke Textbook

Kruschke, Doing Bayesian Data Analysis Kruschke, J. K. (2014). Doing bayesian data analysis, second edition: A tutorial with R, JAGS, and Stan. Academic Press. Excellent textbook – worth the price ($90 from Amazon) Emphasis on classical statistical test problems from a Bayesian perspective. Not so much modeling per se. ♦ Binomial inference problems, anova problems, linear regression problems. Computational Requirements R & JAGS (or OpenBUGS) A programming editor like Rstudio is useful. Psych 548, Miyamoto, Win '16 17 Chapter Outline of Kruschke Textbook

Kruschke, Doing Bayesian Data Analysis Ch 1 – 4: Basic probability background (pretty easy) Ch 5 – 8: Bayesian inference with simple binomial models ♦ Conjugate priors, Gibbs sampling & Metropolis-Hastings algorithm ♦ OpenBUGS or JAGS Ch 9 – 12: Bayesian approach to hierarchical modeling, model comparison, & hypothesis testing. Ch 13: Power & sample size (omit ) Ch 14: Intro generalized linear model Ch 15 – 17: Intro linear regression Ch 18 – 19: Oneway & multifactor anova Ch 20 – 22: Categorical data analysis, logistic regression, probit regression, poisson regression Psych 548, Miyamoto, Win '16 18 Lee & Wagenmakers, Bayesian Graphical Modeling

Psych 548, Miyamoto, Win '16 19 Bayesian Cognitive Modeling Lee, M. D., & Wagenmakers, E. J. (2014). Bayesian cognitive modeling: A practical course. Cambridge University Press. ♦ Michael Lee: ♦ E. J. Wagenmaker: ♦ Equivalent Matlab & R code for book are available at the Psych 548 website and at Lee or Wagenmaker's website. Emphasis is on Bayesian models of psychological processes rather than on methods of data analysis. Lots of examples. Chapters in Lee & Wagenmakers

Table of Contents in Lee & Wagenmakers 20 Psych 548:, Miyamoto, Win ‘16 Computer Setup in CSSCR

Psych 548, Miyamoto, Win '16 21 CSSCR Network & Psych 548 Webpage Click on /Start /Computer. The path & folder name for your Desktop is: C:\users\NetID\Desktop (where "NetID" refers to your NetID) Double click on MyUW on your Desktop. Find Psych 548 under your courses and double click on the Psych 548 website. Download files that are needed for today's class. Save these files to C:\users\NetID\Desktop ♦ Note that Ctrl-D takes you to your Desktop. Run R or RStudio. Psych 548 Website - END Is this information obsolete?

Psych 548 Website Point out where to download the material for today’s class Point out pdf’s for the textbooks. Psych 548, Miyamoto, Win '16 22 NEXT: Time Permitting......

Psych 548, Miyamoto, Win '16 23 General Characteristics of Bayesian Inference The decision maker (DM) is willing to specify the prior probability of the hypotheses of interest. DM can specify the likelihood of the data given each hypothesis. Using Bayes Rule, we infer the probability of the hypotheses given the data Comparison Between Bayesian & Classical Stats - END

How Does Bayesian Stats Differ from Classical Stats? Bayesian: Common Aspects Statistical Models Credible Intervals – sets of parameters that have high posterior probability Bayesian: Divergent Aspects Given data, compute the full posterior probability distribution over all parameters Generally null hypothesis testing is nonsensical. Posterior probabilities are meaningful; p-values are half-assed. MCMC approximations to posterior distributions. Classical: Common Aspects Statistical Models Confidence Intervals – which parameter values are tenable after viewing the data. Classical: Divergent Aspects No prior distributions in general, so this idea is meaningless or self- deluding. Null hypothesis te%sting P-values MCMC approximations are sometimes useful but not for computing posterior distributions. Psych 548, Miyamoto, Win '16 24 Sequential Presentation of the Common & Divergent Aspects

How Does Bayesian Stats Differ from Classical Stats? Bayesian: Common Aspects Statistical Models Credible Intervals – sets of parameters that have high posterior probability Bayesian: Divergent Aspects Given data, compute the full posterior probability distribution over all parameters Generally null hypothesis testing is nonsensical. Posterior probabilities are meaningful; p-values are half-assed. MCMC approximations to posterior distributions. Classical: Common Aspects Statistical Models Confidence Intervals – which parameter values are tenable after viewing the data. Classical: Divergent Aspects No prior distributions in general, so this idea is meaningless or self- deluding. Null hypothesis te%sting P-values MCMC approximations are sometimes useful but not for computing posterior distributions. Psych 548, Miyamoto, Win '16 25 END

How Does Bayesian Stats Differ from Classical Stats? Bayesian: Common Aspects Statistical Models Credible Intervals – sets of parameters that have high posterior probability Bayesian: Divergent Aspects Given data, compute the full posterior probability distribution over all parameters Generally null hypothesis testing is nonsensical. Posterior probabilities are meaningful; p-values are half-assed. MCMC approximations to posterior distributions. Classical: Common Aspects Statistical Models Confidence Intervals – which parameter values are tenable after viewing the data. Classical: Divergent Aspects No prior distributions in general, so this idea is meaningless or self-deluding. Null hypothesis testing P-values MCMC approximations are sometimes useful but not for computing posterior distributions. Psych 548, Miyamoto, Win '16 26 END

How Does Bayesian Stats Differ from Classical Stats? Bayesian: Common Aspects Statistical Models Credible Intervals – sets of parameters that have high posterior probability Bayesian: Divergent Aspects Given data, compute the full posterior probability distribution over all parameters Generally null hypothesis testing is nonsensical. Posterior probabilities are meaningful; p-values are half-assed. MCMC approximations to posterior distributions. Classical: Common Aspects Statistical Models Confidence Intervals – which parameter values are tenable after viewing the data. Classical: Divergent Aspects No prior distributions in general, so this idea is meaningless or self- deluding. Null hypothesis testing P-values MCMC approximations are sometimes useful but not for computing posterior distributions. Psych 548, Miyamoto, Win '16 27 END