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<!DOCTYPE html>
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8"/>

<title>Odds and probabilities using BayesFactor</title>

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" alt="BayesFactor logo"/></p>

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<h1>Odds and probabilities using BayesFactor</h1>

<h2>Richard D. Morey</h2>

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<hr/>

<p>The Bayes factor is only one part of Bayesian model comparison. The Bayes factor represents the relative evidence between two models &ndash; that is, the change in the model odds due to the data &ndash; but the odds are what are being changed. For any two models \({\cal M}_0\) and \({\cal M}_1\) and data \(y\),</p>

<p>\[
\frac{P({\cal M}_1\mid y)}{P({\cal M}_0\mid y)} = \frac{P(y \mid {\cal M}_1)}{P(y\mid{\cal M}_0)} \times\frac{P({\cal M}_1)}{P({\cal M}_0)};
\]
that is, the posterior odds are equal to the Bayes factor times the prior odds.</p>

<p>Further, these odds can be converted to probabilities, if we assume that all the models sum to known probability.</p>

<h3>Prior odds with BayesFactor</h3>

<pre><code class="r">data(puzzles)
bf = anovaBF(RT ~ shape*color + ID, whichRandom = &quot;ID&quot;, data = puzzles)
bf
</code></pre>

<pre><code>## Bayes factor analysis
## --------------
## [1] shape + ID                       : 2.89 ±1.63%
## [2] color + ID                       : 2.8  ±0.85%
## [3] shape + color + ID               : 11.6 ±1.48%
## [4] shape + color + shape:color + ID : 4.24 ±3.1%
## 
## Against denominator:
##   RT ~ ID 
## ---
## Bayes factor type: BFlinearModel, JZS
</code></pre>

<p>With the addition of <code>BFodds</code> objects, we can compute prior and posterior odds. A prior odds object can be created from the structure of an existing BayesFactor object:</p>

<pre><code class="r">prior.odds = newPriorOdds(bf, type = &quot;equal&quot;)
prior.odds
</code></pre>

<pre><code>## Prior odds
## --------------
## [1] shape + ID                       : 1
## [2] color + ID                       : 1
## [3] shape + color + ID               : 1
## [4] shape + color + shape:color + ID : 1
## 
## Against denominator:
##   RT ~ ID 
## ---
## Model type: BFlinearModel, JZS
</code></pre>

<p>For now, the only type of prior odds is &ldquo;equal&rdquo;. However, we can change the prior odds to whatever we like with the <code>priorOdds</code> function:</p>

<pre><code class="r">priorOdds(prior.odds) &lt;- c(4,3,2,1)
prior.odds
</code></pre>

<pre><code>## Prior odds
## --------------
## [1] shape + ID                       : 4
## [2] color + ID                       : 3
## [3] shape + color + ID               : 2
## [4] shape + color + shape:color + ID : 1
## 
## Against denominator:
##   RT ~ ID 
## ---
## Model type: BFlinearModel, JZS
</code></pre>

<h3>Posterior odds with BayesFactor</h3>

<p>We can multiply the prior odds by the Bayes factor to obtain posterior odds:</p>

<pre><code class="r">post.odds = prior.odds * bf
post.odds
</code></pre>

<pre><code>## Posterior odds
## --------------
## [1] shape + ID                       : 11.6 ±1.63%
## [2] color + ID                       : 8.4  ±0.85%
## [3] shape + color + ID               : 23.2 ±1.48%
## [4] shape + color + shape:color + ID : 4.24 ±3.1%
## 
## Against denominator:
##   RT ~ ID 
## ---
## Model type: BFlinearModel, JZS
</code></pre>

<h3>Prior/posterior probabilities with BayesFactor</h3>

<p>Odds objects can be converted to probabilities:</p>

<pre><code class="r">post.prob = as.BFprobability(post.odds)
post.prob
</code></pre>

<pre><code>## Posterior probabilities
## --------------
## [1] shape + ID                       : 0.239  ±NA%
## [2] color + ID                       : 0.174  ±NA%
## [3] shape + color + ID               : 0.479  ±NA%
## [4] shape + color + shape:color + ID : 0.0875 ±NA%
## [5] ID                               : 0.0207 ±NA%
## 
## Normalized probability:  1  
## ---
## Model type: BFlinearModel, JZS
</code></pre>

<p>By default the probabilities sum to 1, but we can change this by renormalizing. Note that this normalizing constant is arbitrary, but it can be helpful to set it to specific values.</p>

<pre><code class="r">post.prob / .5
</code></pre>

<pre><code>## Posterior probabilities
## --------------
## [1] shape + ID                       : 0.12   ±NA%
## [2] color + ID                       : 0.0868 ±NA%
## [3] shape + color + ID               : 0.24   ±NA%
## [4] shape + color + shape:color + ID : 0.0438 ±NA%
## [5] ID                               : 0.0103 ±NA%
## 
## Normalized probability:  0.5  
## ---
## Model type: BFlinearModel, JZS
</code></pre>

<p>In addition, we can select subsets of the probabilities, and the normalizing constant is adjusted to the sum of the model probabilities:</p>

<pre><code class="r">post.prob[1:3]
</code></pre>

<pre><code>## Posterior probabilities
## --------------
## [1] shape + ID         : 0.239 ±NA%
## [2] color + ID         : 0.174 ±NA%
## [3] shape + color + ID : 0.479 ±NA%
## 
## Normalized probability:  0.892  
## ---
## Model type: BFlinearModel, JZS
</code></pre>

<p>&hellip;which can, in turn, be renormalized:</p>

<pre><code class="r">post.prob[1:3] / 1
</code></pre>

<pre><code>## Posterior probabilities
## --------------
## [1] shape + ID         : 0.268 ±NA%
## [2] color + ID         : 0.195 ±NA%
## [3] shape + color + ID : 0.537 ±NA%
## 
## Normalized probability:  1  
## ---
## Model type: BFlinearModel, JZS
</code></pre>

<p>In the future, the ability to filter these objects will be added, as well as model averaging based on posterior probabilities and samples.</p>

<hr/>

<p>Social media icons by <a href="http://www.awicons.com/">Lokas Software</a>.</p>

<p><em>This document was compiled with version 0.9.12-2 of BayesFactor (R Under development (unstable) (2015-09-14 r69389) on x86_64-apple-darwin13.4.0).</em></p>

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