R scale Cauchy prior

ssk · September 2016

Hello,

I am teaching a statistics class and while the examinable contents are based on the frequentist approach, I wish to briefly show my students an alternative way of analysing data using Bayesian statistics and I am using JASP to do this. Therefore, I would like to know a little bit more on the Cauchy prior width (r scale) for the Bayesian t-test. Is it correct for me to say that the default value of r = 0.707 translates to an expectation of an effect size of d = 0.707? In other words, probability mass corresponding to effect magnitude less than 0.707 will be 0.5 and more than 0.707 will be 0.5 as well?

Additionally, where does this default prior width come from?

Thank you for your time.

EJ · September 2016

Hi ssk,

The best way to understand what the width parameter does is to try a few values and inspect the prior-posterior plot. In the current version of JASP, all Cauchy priors are centered on 0, so the median is 0 (the distribution does not have a mean -- yes, this is counterintuitive). Richard Morey usually explains the r parameter as follows. Higher r values mean a wider distribution. Specifically, when r=x this means that 50% of the values lie between -x and x. So when r=.707, 50% of the values lie between -0.707 and +0.707 (the interquartile range). Some people find this a pretty wide range, but on the other hand, most mass is assigned to values near zero. This particular default width is really 0.5*sqrt(2). It was chosen to be reasonable (the original proposal from Jeffreys was r=1) and the sqrt(2) comes in because of the extension to the ANOVA (see the more complicated work by Rouder and Morey on Bayesian ANOVA)

Cheers,
E.J.

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