Bayesian Inference Using JASP Eric-Jan Wagenmakers 1

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Outline Bayesian inference Bayesian parameter estimation Bayesian hypothesis testing The Bayesian t-test Example: Turning the hands of time 3

Bayesian Inference in a Nutshell In Bayesian inference, uncertainty or degree of belief is quantified by probability. Prior beliefs are updated by means of the data to yield posterior beliefs. 4

Outline Bayesian inference Bayesian parameter estimation Bayesian hypothesis testing The Bayesian t-test Example: Turning the hands of time 5

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Outline Bayesian inference Bayesian parameter estimation Bayesian hypothesis testing The Bayesian t-test Example: Turning the hands of time 16

Bayesian Hypothesis Test Suppose we have two models, H0 and H1. Which model is better supported by the data? The model that predicted the data best! The ratio of predictive performance is known as the Bayes factor (Jeffreys, 1961).

Bayesian Hypothesis Tests

Bayesian Hypothesis Tests

Bayesian Hypothesis Tests

Bayesian Hypothesis Tests

Bayes Factor Inverse

Bayes Factor Transitivity

Bayes Factor Transitivity

Guidelines for Interpretation of the Bayes Factor BF Evidence 1 – 3 Anecdotal 3 – 10 Moderate 10 – 30 Strong 30 – 100 Very strong >100 Extreme 25

Visual Interpretation of the Bayes Factor 26

Visual Interpretation of the Bayes Factor 27

Visual Interpretation of the Bayes Factor 28

Advantages of the Bayes Factor Quantifies evidence instead of forcing an all- or-none decision. Allows evidence to be monitored as data accumulate. Able to distinguish between “data support H0” and “data are not diagnostic”.

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Disadvantages of the Bayes Factor Where are the course books for psychologists? Where is the software?

First Annual JASP Workshop A Fresh Way to do Bayesian Statistics jasp-stats.org August 6 & August 7, 2015 University of Amsterdam

Fifth Annual JAGS and WinBUGS Workshop Bayesian Modeling for Cognitive Science http://bayescourse.socsci.uva.nl/ August 10 - August 14, 2015 University of Amsterdam

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Outline Bayesian inference Bayesian parameter estimation Bayesian hypothesis testing The Bayesian t-test Example: Turning the hands of time 36

Bayes Factor for the t Test Predictive Success for the Null Hypothesis Predictive Success for the Alternative Hypothesis 37

Bayes Factor for the t Test Prob. of Data Under the Null Hypothesis Prob. of Data Under the Alternative Hypothesis H0 states that effect size δ = 0. But how do we specify H1? 38

is the weighted average of the likelihood ratios. Effect size δ under H1 Likelihood ratio p(data | H0) p(data | δ = .1) The Bayes factor is the weighted average of the likelihood ratios. The weights are given by the prior plausibility assigned to the effect sizes. δ = .1 δ = .3 δ = .5 • p(data | H0) p(data | δ = .3) p(data | H0) p(data | δ = .5) 39

So we need to assign weight to the different values of effect size. These weigths reflect the relative plausibility of the effect sizes before seeing the data. For complicated reasons, a popular default choice is to assume a Cauchy distribution (like a Normal, but with fatter tails): 40

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Outline Bayesian inference Bayesian parameter estimation Bayesian hypothesis testing The Bayesian t-test Example: Turning the hands of time 42

Turning the Wheels of Time Topolinski and Sparenberg (2012): clockwise movements induce psychological states of temporal progression and an orientation toward the future and novelty. Concretely: participants who turn kitchen rolls clockwise report more openness to experience.

Turning the Wheels of Time

Turning the Wheels of Time Let's demonstrate: How to run a Bayesian t-test in JASP; How to interpret the output; How to conduct a sequential analysis; How to assess the robustness of the result.

Conclusions Bayesian hypothesis tests have a number of practical advantages; These advantages are easily available through JASP; The example discussed here was simple, but JASP handles more complicated analyses as well!