Psych 548, Miyamoto, Win '15 1 Set Up for Students Your computer should already be turned on and logged in. Open a browser to the Psych 548 website ( you.

1Psych 548, Miyamoto, Win '15

Set Up for Students• Your computer should already be turned on and logged in.

• Open a browser to the Psych 548 website (you can get it from MyUW)

http://faculty.washington.edu/jmiyamot/p548/p548-set.htm

• Download the zip file: p548.zip .

Unzip the zip file to C:\temp . This process will create a

subdirectory, C:\temp\p548 . The files for today’s class are in this

directory or one of its subdirectories.

• Run R. Run Rstudio. Load any pdf handouts for today into

Acrobat.

Bayesian Statistics, Modeling & Reasoning

What is this course about?

Psychology 548

Bayesian Statistics, Modeling & Reasoning

Instructor: John Miyamoto

01/05/2015: Lecture 1-1

This Powerpoint presentation may contain macros that were used to create the slides. The macros aren’t needed to view the slides. If necessary, you can disable the macros without any change to the presentation.

Outline

• What is Bayesian inference?

• Why is Bayesian statistics, modeling & reasoning relevant to psychology?

• What is Psych 548 about?

• Familiarize students with the set up for using MGH 058

• Explain Psych 548 website

• Intro to R

• Intro to RStudio

• Intro to the R to BUGS interface

Psych 548, Miyamoto, Win '15 3

Lecture probably ends here

4

Bayes Rule – What Is It?

• Reverend Thomas Bayes, 1702 – 1761English Protestant minister & mathematician

• Bayes Rule is fundamentally important to:♦ Bayesian statistics♦ Bayesian decision theory♦ Bayesian models in psychology

Psych 548, Miyamoto, Win '15

P Data|Hypothesis P(Hypothesis)P(Hypothesis|Data)

P(Data)=

n

i ii 1

P(Data) P Data | Hypothesis P Hypothesis

Bayes Rule – Why Is It Important?


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 conceptso Concept 1: Bayesian inference is a model of optimal learning from

experience.

o Concept 2: Bayesian decision theory describes optimal strategies for taking actions in an uncertain environment. Optimal gambling.

o 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


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?o 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


Four Roles for Bayesian Reasoning in Psychology

1. Bayesian statistics: Analyzing datao E.g., is the slope of the regression of grades on IQ the same for boys as for

girls?o E.g., are there group differences in an analysis of variance?

Four Roles …. (Continued)



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)o Judgment and decision making – This is a major issue.o Human causal reasoning – is it Bayesian or quasi-Bayesian?o Modeling neural decision making – many proposed models have a strong

Bayesian flavor.

Four Roles …. (Continued)



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)

Psych 548:

Focus on Topics (1) and (3).

Includes a little bit of (4).

Graphical Representation of Psych 548 Focus on Stats/Modeling


Graphical Representation of Psych 548

Bayesian Statistics& Modeling:

R, OpenBUGS,

JAGS

Bayesian Models in Child & Adult Psychology & Neuroscience

Psych 548

Graph & Text Showing the History of S, S-Plus & R


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)

o currently the standard computing framework for statisticians worldwideMany contributors to its development

o Excellent general computation. Powerful & flexible.

o Great graphics.o Multiplatform: Unix, Linux, Windows, Maco User must like programming

BUGS, WinBUGS, OpenBUGS, JAGS

S

S-PlusR

Ancestry of R


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)o Windows only

• OpenBUGS – Open source (circa 2006) o Mainly Windows. Runs within a virtual Windows machine on a Mac.

• JAGS – Open source (circa 2007)o 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.


Basic Structure of Bayesian Computation

R

data preparation

analysis of results

JAGS

Computes approximation to the posterior distribution.Includes diagnostics.

rjags functions

rjags functions

rjagsrunjags

OpenBUGS/WinBUGS/

StanRBRugs functions

Brugs functions

BRugsR2WinBUGS

rstan

Outline of Remainder of the Lecture: Course Outline & General Information

RStudio

• Run RStudio

• Run R from within RStudio



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

5 Minute Break

• Introduce selves upon return

Psych 548, Miyamoto, Win '15 16Course Goals


Course Goals

• Learn the theoretical framework of Bayesian inference.

• Achieve competence with R, OpenBUGS and JAGS.

• Learn basic Bayesian statisticso Learn how to think about statistical inference from a Bayesian standpoint. o Learn how to interpret the results of a Bayesian analysis. o Learn basic tools of Bayesian statistical inference - testing for convergence,

making standard plots, examing samples from a posterior distribution.

---------------------------------------------------------------

Secondary Goalso Bayesian modeling in psychology

o Understand arguments about Bayesian reasoning in the psychology of reasoning. The pros and cons of the heuristics & biases movement.

Kruschke Textbook

Main Text: 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 '15 18Chapter Outline of Kruschke Textbook

Main Text: 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 '15 19Lee & Wagenmakers, Bayesian Graphical Modeling


Workbook on Bayesian Graphical Modeling

Kruschke, J. K. (2014). Doing bayesian data analysis, second

edition: A tutorial with R, JAGS, and Stan. Academic Press.o Michael Lee: http://www.socsci.uci.edu/~mdlee/bgm.html o E. J. Wagenmaker: http://users.fmg.uva.nl/ewagenmakers/BayesCourse/BayesBook.html

• 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 theory. Lots of examples.

Computer Setup in CSSCR


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 o Note that Ctrl-D takes you to your Desktop.

• Run R.

• Run RStudio. Psych 548 Website - END

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 '15 22END

Time Permitting: Proceed to Bayes Rule


24

Bayes Rule

• Reverend Thomas Bayes, 1702 – 1761British Protestant minister & mathematician

• Bayes Rule is fundamentally important to:♦ Bayesian statistics♦ Bayesian decision theory♦ Bayesian models in psychology



P(Data)=

n

i ii 1

P(Data) P Data | Hypothesis P Hypothesis

Next: Explanation of Bayes Rule

25

Bayes Rule – Explanation



P(Data)=

Odds Form of Bayes Rule

Posterior Probability

of the Hypothesis

Likelihood of the Data

Prior Probability of

the Hypothesis

NormalizingConstant

26

Bayes Rule – Explanation



P(Data)=


Posterior Probability

of the Hypothesis

Likelihood of the Data

Prior Probability of

the Hypothesis

NormalizingConstant

27

Bayes Rule – Odds Form

P D | H P(H)P H | D

P D


P D | H P(H)P H | D

P D

P H | D

P(H | D)

P D | H P(H)

P D | H P(H)

Bayes Rule for H given D

Bayes Rule for not-H given D


Explanation of Odds form of Bayes Rule

28

Bayes Rule (Odds Form)

H = a hypothesis, e.g.., hypothesis that the patient has cancer

= the negation of the hypothesis, e.g.., the hypothesis that the patient does not have cancer

D = the data, e.g., a + result for a cancer test


P H | D

P(H | D)

P D | H P(H)

P D | H P(H)

Posterior Odds

Likelihood Ratio(diagnosticity)

Prior Odds(base rate)

H

Interpretation of a Medical Test Result


Bayesian Analysis of a Medical Test Result(Look at Handout)

QUESTION: A physician knows from past experience in his practice

that 1% of his patients have cancer (of a specific type) and 99%

of his patients do not have the cancer. He also knows the

probabilities of a positive test result (+ result) given cancer and

given no cancer. These probabilities are:

P(+ test | Cancer) = .792 and P(+ test | no cancer) = .096

Suppose Mr. X has a positive test result.

What is the probability that Mr. X has cancer?

• Write down your intuitive answer. (Note to JM: Write estimates on board)

Solution to this problem


Given Information in the Diagnostic Inference from a Medical Test Result

• P(+ test | Cancer) = .792 (true positive rate a.k.a. hit rate)

• P(+ test | no cancer) = .096 (false positive rate a.k.a. false alarm rate)

• P(Cancer) = Prior probability of cancer = .01

• P(No Cancer) = Prior probability of no cancer

= 1 - P(Cancer) = .99

• Mr. X has a + test result.

What is the probability that Mr. X has cancer?

Solution to this problem


Bayesian Analysis of a Medical Test Result

P(+ test | Cancer) = 0.792 and P(+ test | no cancer) = 0.096

P(Cancer) = Prior probability of cancer = 0.01

P(No Cancer) = Prior probability of no cancer = 0.99

P(Cancer | + test) = 1 / (12 + 1) = 0.077

Digression concerning What Are Odds?

P cancer | test P test | cancer P(cancer)P no cancer | test P test | no cancer P(no cancer)

0.792 0.010.0833 1/12

0.096 0.99


Digression: Converting Odds to Probabilities

• If X / (1 – X) = Y

• Then X = Y(1 – X) = Y – XY

• So X + XY = Y

• So X(1 + Y) = Y

• So X = Y / (1 + Y)

• Conclusion: If Y are the odds for an event,

then, Y / (1 + Y) is the probability of the event

Return to Slide re Medical Test Inference


Bayesian Analysis of a Medical Test Result

P(+ test | Cancer) = 0.792 and P(+ test | no cancer) = 0.096

P(Cancer) = Prior probability of cancer = 0.01

P(No Cancer) = Prior probability of no cancer = 0.99

P(Cancer | + test) = (1/12) / (1 + 1/12) = 1 / (12 + 1)

= 0.077

Compare the Normative Result to Physician’s Judgments

P cancer | test P test | cancer P(cancer)P no cancer | test P test | no cancer P(no cancer)

0.792 0.010.0833 1/12

0.096 0.99


Continue with the Medical Test Problem

• P(Cancer | + Result) = (.792)(.01)/(.103) = .077

• Posterior odds against cancer are (.077)/(1 - .077) or

about 1 chance in 12.

Notice: The test is very diagnostic but still P(cancer | + result) is

low because the base rate is low.

• David Eddy found that about 95 out of 100 physicians stated that

P(cancer | +result) is about 75% in this case (very close

to the 79% likelihood of a + result given cancer).

General Characteristics of Bayesian Inference


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 testing

• P-values

• MCMC approximations are sometimes useful but not for computing posterior distributions.

Psych 548, Miyamoto, Win '15 36END

Psych 548, Miyamoto, Win '15 1 Set Up for Students Your computer should already be turned on and logged in. Open a browser to the Psych 548 website ( you.

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