Bayesian Inference Using JASP

Eric-Jan

Wagenmakers

Outline

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

Bayesian Inferencein 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.

Outline

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

Outline

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

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 Moderate10 – 30 Strong 30 – 100 Very strong >100 Extreme

Visual Interpretation of the Bayes Factor

Visual Interpretation of the Bayes Factor

Visual Interpretation of the Bayes Factor

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”.

Disadvantages of the Bayes Factor

Where are the course books for psychologists?

Where is the software?

August 6 & August 7, 2015University of Amsterdam

First Annual JASP WorkshopA Fresh Way to do Bayesian Statistics

jasp-stats.org

August 10 - August 14, 2015University of Amsterdam

Fifth Annual JAGS and WinBUGS WorkshopBayesian Modeling for Cognitive Science

http://bayescourse.socsci.uva.nl/

Outline

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

Bayes Factor for the t Test

Predictive Success for the Null Hypothesis

Predictive Success for the Alternative Hypothesis

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?

Effect size δ under H1

δ = .1

δ = .3

δ = .5

•••

•••

•••

Likelihood ratio

p(data | H0)

p(data | δ = .1)

p(data | H0)

p(data | δ = .3)

p(data | H0)

p(data | δ = .5)

The Bayes factoris the weighted average of the likelihood ratios.The weights are given by the prior plausibility assigned to the effect sizes.

So we need to assign weight to the different values of effect size.

For complicated reasons, a popular default choice is to assume a Cauchy distribution (like a Normal, but with fatter tails):

These weigths reflect the relative plausibility of the effect sizes before seeing the data.

Outline

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

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!