Foreman-Mackey and I sprinted on The Thresher in the hopes of giving me something to say in my Hauskolloquium here at MPIA tomorrow. We figured out something (already known) about non-negative priors: They can lead to severe biases. Consider, for example, PSF fitting with non-negative priors. If there is any noise in the system at the outkirts of the data used to find or fit the PSF, a system with non-negative priors can only capture the positive excursions; it can't also capture the negative excursions. This leads to two kinds of biases: One is that the PSF inferred with non-negative priors is always fatter than the PSF you would obtain by normal linear least squares. The other is that the PSF never really goes exactly to zero in the wings; it just can't. These problems arise, in my view, because we are doing point estimation; if we were carrying forward a full posterior PDF for the PSF we would be fine, but we just don't yet know how to do that! We came up with short-term hacks to deal with our problems and get me ready for my talk tomorrow, which is about extracting information from collections of images.
Interesting. Certainly naive non-negative priors can do really bad things. The first time I tried it, my implied prior on total flux was so strong that it swamped the data and told me the whole sky was actually bright. This wasn't with a point estimate either, but with full sampling!
ReplyDeleteI would bet that in the long run the best results will come from non-negative priors, but the prior needs to be designed cleverly.