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Introduction

PSYC 573

University of Southern California

January 11, 2022

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History of Bayesian Statistics

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  • 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

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, Wikimedia Commons

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)1

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

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Resurrection

  • 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

Why Should You Learn About the Bayesian Way?

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

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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

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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?

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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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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

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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: (−∞,∞) (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) × 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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Accepting the Null

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

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History of Bayesian Statistics

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