Beta prior widths

MattiHeino

Hi!

I've understood (perhaps falsely?), that a beta prior width of 1 assigns equal prior probability to all correlation values between -1 and 1. Does a beta prior width of 0.5 thus indicate that values more extreme than |0.5| are considered impossible, or does the logic follow the beta distribution somehow?

Many thanks,

Matti

EJ

The logic follows from the (streched) beta. I recall that the prior width of 0.5 generates a beta(2,2) stretched from -1 to 1 (because 2 = 1/0.5; the width is inversely related to the parameters of the beta). When you run an analysis you can tick the box that plots the prior -- this allows you to see what happens.
Cheers,
E.J.

MattiHeino

Great, thanks!

MSB

Hi, I have a similar question about beta prior width.

I have a measurement that I know cannot have a higher correlation than 0.6 (due to a known low reliability).
How do I know what to set my beta prior width according to this prior knowledge?

Thanks,
M

EJ

Hi M,
Right now JASP does not do truncation on the beta prior. However, you can take the results and add the truncation after the fact. Here is how it works. JASP gives you the Bayes factor BF10 of a H1 model versus H0. You can then use a "trick" to obtain the Bayes factor of that same H1 against the version that implements the truncation, BF1t. The trick is to compare the prior proportion consistent with the restrictions to the posterior proportion consistent with the restriction. With both Bayes factors in hand, the desired result can be obtained by transitivity: BFt0 = BF10 / BF1t. It is explained here: http://www.ejwagenmakers.com/2014/MoreyWagenmakers2014.pdf. You can obtain the required prior and posterior proportions from the fact that both are beta distributions.

Bottom line: JASP does not do this yet, but you can do it yourself if you really need to.

Cheers,
E.J.

MSB

Hi EJ,
Thanks for the quick response!

I'm afraid I haven't quite understood the meaning of Beta prior widths, of how to set them...

In accordance with your "trick", if I want to truncation my H0 to a max of r=0.6, would I set my Beta prior width=0.6*2? This would then yield a BF1t?

Thanks again!
M

EJ

Hi M,

You can view the effect of the Beta* prior widths by trying out a few values and ticking the plot option to view the results. Basically, the standard beta distribution is defined for a parameter ranging from 0 to 1 (check out Wikipedia for an illustration of different shapes). We are dealing with a correlation, and so we stretch the beta to cover the interval from -1 to 1. OK. Now the default choice is Beta(1,1): the flat prior. If you change the prior width to, say, .5, you will get a stretched Beta(2,2): so the prior width is 1/a, with the corresponding stretched Beta prior being Beta(a,a). The lower the prior width, the more peaked the Beta distribution will be around 0.

Now none of this does truncation. If you want truncation then you'll have to use the results that JASP gives you and do some extra work. The stretched beta priors in JASP range from -1 to 1 (or from -1 to 0, or from 0 to 1, if you use a one-sided test) -- so no truncation is possible in JASP itself right now.

Cheers,
E.J.

MSB

Hi EJ,

Thank again for the detailed response.
I think I got it - Thanks!

M