Introduction

# Introduction
## PSYC 573
### University of Southern California
### January 11, 2022

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# Warmup Quizz

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# Scoring

1. a = 1 point, b = 3 points, c = 2 points; 
2. a = 1 point, b = 3 points, c = 1 point; 
3. a = 3 points, b = 1 point; 
4. a = 3 points, b = 1 point.

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# Scoring

- 4--5: your current thinking is fairly frequentist
- 9--12: alignment with the Bayesian philosophy
- 6--8: strengths in both philosophies

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From https://www.bayesrulesbook.com/chapter-1.html

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# History of Bayesian Statistics

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- Video intro: https://www.youtube.com/watch?v=BcvLAw-JRss

- A nice popular science book by Sharon Bertsch McGrayne: *The theory that would not die*

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# Historical Figures

Thomas Bayes (1701--1762)

- English Presbyterian minister
- "An Essay towards solving a Problem in the Doctrine of Chances", edited by Richard Price after Bayes's death

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Pierre-Simon Laplace (1749--1827)

- French Mathematician
- Formalize Bayesian interpretation of probability, and most of the machinery for Bayesian statistics

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Image credit: [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Thomas_Bayes.gif), [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Pierre-Simon_Laplace.jpg)

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# In the 20th Century

- Bayesian---way to do statistics until early 1920s

- Ronald Fisher and Frequentist scholars took over 
    * "The theory of inverse probability is founded upon an error, and must be wholly rejected" (Fisher, 1925, p. 10)<sup>1</sup>

[1]: Aldrich, J. (2008). R. A. Fisher on Bayes and Bayes' theorem. *Bayesian Analysis, 3*(1), 161--170.

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# Resurrection

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- Alan Turing's algorithms in code breaking in World War II

- *Markov Chain Monte Carlo* (MCMC) algorithm
    * Bring Bayesian back to the main stream of statistics
    
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Image credit: [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Alan_Turing_Aged_16.jpg)

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## Why Should You Learn About the Bayesian Way?

- Gigerenzer (2004): It is one tool of your statistical toolbox

- Increasingly used as alternative to frequentist statistics

- Computationally more stable for complex models

- A coherent way of incorporating prior information
    * Common sense knowledge, previous literature, sequential experiments, etc

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# Bayesian Idea 1

### Reallocation of credibility across possibilities

Hypothetical example: How effective is a vaccine?

Prior (before collecting data)

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# Updating Beliefs

After seeing results of a trial
- 4/5 with the vaccince improved
- 2/5 without the vaccine improved

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# Possibilities = Parameter Values

- Parameter: Effectiveness of the vaccine
- Possibilities: Not effective, mildly effective, very effective

> Here the parameter is a discrete variable

- Parameter: Risk reduction by taking the vaccine
- Possibilities: `\((-\infty, \infty)\)` (Any real number)

> Here the parameter is a continuous variable

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Using Bayesian analysis, one obtains updated/**posterior probability** for every possibility of a parameter, given the **prior** belief and the **data**

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# Steps of Bayesian Data Analysis

"Turning the Bayesian crank"

1. Identify data
2. Define a mathematical model with parameters
3. Specify priors on parameters
4. Obtain and interpret posterior distributions of the parameters
5. Posterior predictive check

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# Example

Frank et al. (2019, Cognition and Emotion)

- Response time for 2 (Dutch--native vs. English--foreign) `\(\times\)` 2 (lie vs. truth) experimental conditions

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# Posterior of Mean RTs by Conditions

L = Lie, T = Truth; D = Dutch, E = English

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# From Priors to Posteriors

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### Accepting the Null

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### Posterior Predictive Check

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# Multiple Experiments

Kay, Nelson, & Hekler (2016, p. 4525, https://dl.acm.org/doi/abs/10.1145/2858036.2858465)

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# Syllabus

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# Homework 1