Cauchy prior widths
Hi!
I just read Will Gervais's blog post* about bayes factors, where he uses JASP (as well). In the comments he says he thinks that for nondirectional tests, "each half of the 'bell' extending up and down from zero will be centered at the 'Cauchy prior width' value."
Is this the correct interpretation, just to confirm? I'm also wondering, wouldn't I want to place the prior mean at d=0.3 for example?
Thanks!
Matti
*http://willgervais.com/blog/2015/11/20/playing-with-bayes-factors
ps. took me a LONG time to locate the "help" button in JASP; maybe it's just me being blind to small print in the top right corner 
Comments
Hi Matti,
Good points.
(1) I have added "make help button more prominent" as a feature request via the JASP GitHub system. By the way, this is easy to do: just go to jasp-stats.org, to "development" and then to "feature requests".
(2) With respect to the prior, right now all nondirectional (two-sided) priors in JASP are centered on zero, as is standard practice in Bayesian statistics. So currently it is not possible in JASP to specify a symmetric prior centered on, say, 0.3. The nature of the prior will be evident when you plot the prior and posterior distribution (which I strongly recommend in general).
Now I agree that sometimes you might want to center the prior on a nonzero value. For instance, this is what we do for the "replication Bayes factor" (http://www.ejwagenmakers.com/2014/VerhagenWagenmakers2014.pdf). We are currently working to add these kinds of tests to JASP, and I believe Richard is looking into this for the BayesFactor package as well (we strive to keep the programs similar in functionality).
(3) For a symmetric Cauchy prior, the width equals the interquartile range. So when r=.707 you are 50% confident that the true effect size will lie between -.707 and .707. Richard can correct me if I'm wrong here. Some people feel such a setting is too wide, and they want to use lower values of r. However, note that if you use Cauchy priors centered on 0 (as we currently do in JASP), lowering r will make H1 very similar to H0, and almost all tests become uninformative. So when people are worried that H1 is disadvantaged because of its optimistic predictions, they should realize that H0 is also disadvantaged: H1 has most of its mass around 0. One way around this is to allow nonzero mean for the priors; in that case, it does make sense to set r to a low value. In my experience, most of these issues don't lead to qualitatively different results. The difference between p-values and Bayes factors is much bigger than that between different Bayes factors with plausible values for the prior distribution.
(4) I generally like directional tests; they better represent the hypotheses under scrutiny.
Cheers,
E.J.
Many thanks!
This was extremely useful. Also, I just found where to activate e-mail notifications on this board, so probably less answer lag when I'll be back
Best wishes,
Matti
I also found the above explanation very useful. Can you explain why centering the prior on 0 is the standard/preferred practice with Bayesian analyses vs centering on some other value? Under what conditions would it be important to not center on 0?
Standard practice is to center on zero. I guess the idea is originally from Jeffreys, who viewed 0 as the "general law" or "invariance", and then allowed H1 to depart from that value. If you don't center on zero you can have a more diagnostic test (because H0 and H1 start to differ in their predictions more).