yet another generative model for images

Work continued today with Frean and Brewer. After a second long day of talk, we came up with a plan, which is a generative model for astronomical images and (limited) implementation goals. The model is that the image has compact patches generated by a "source" model and the rest is generated by a "background" model. Each of these submodels—every source and the background—has a unique generating pixel histogram; no other spatial information in the image is used at all! That is, the model is that there are contiguous patches, within each of which the pixel data are produced by iid draws from a unique histogram model.

The nice thing is that Frean has good "Dirichlet multinomial" technology for marginalizing over histograms under a particular prior, so the model can be marginalized trivially. This project violates many of my rules, because it makes very little use of my knowledge of astronomy, and it employs conjugate priors, which are always wrong. That said, all models are wrong, so sometimes expediency is a good idea. This project is very aligned with others of my rules, because we have been able to write down a full likelihood and priors over all of the nuisance parameters. That is, we have a generative model for astronomical images. It isn't a good model, but it is (or might be) very fast to compute. We also have a graphical-model representation of it all.

We are ready to code, and hope to start tomorrow. One strong intuition we have is that there are very simple index-algebra and precomputation things we can do to make the (marginalized) likelihood values very easy to compute. I hope this ends up being true!

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