I know that for the cholesky correlation matrix - eta =1 is considered an uninformative prior.
However, when I generate samples with this prior (see attachment), it seems that the median correlation values in the non diagonals is close to .40! in most disciplines those would be quite significant correlations, meaning our prior assumes strong correlations.
it seems that something like eta=23 which generates a median correlation of .1 (at the border of being a meaningful correlation) is more suitable.
Is there something i’m missing/misunderstanding? Please see the attached notebook which specifies how I calculated those numbers.
It’s uniform over the space of positive definite correlation matrices. In your plot of eta=1 you see that this is clearly not uniform across all values of correlation. The reason for this is that the space of PD correlation matrices concentrates around 0 as the dimension of the correlation matrix increases. Geometrically this is like all the volume is contained on the surface of the hypersphere.
You have some a prior information and so should incorporate it. The LKJ prior is great when you have no information about any of the correlations or the information you do have is on all of the correlations and not any one specifically. However, if you don’t want the bell shaped distribution when constraining the median correlation to be 0.1 or whatever then you have to do some different things. You can look at adding a bound like 0.3 and messing around with eta, putting bounds on is described at Constraints on LKJ prior - #12 by spinkney. That also allows you to constrain different correlations to be equal, which also includes making them into blocks that all are the same value.
If you suspect some correlations spike towards the pole (-1, 1) but many are small or 0 but you don’t know which. Well, there’s not too much to do today. I’m working on priors for this but my day job gets in the way of progress.
thanks so much spinkney!- this distribution shape suits me - i dont have any particilar strong assumptions, just want to be more conservative in the correlations i expect. so you dont see anything potentially problematic with using an eta as high as 23? seems better than forcing an artificial bound like .3, no?