cosmology, now with less approximation; RNADE

In the morning, Mike O'Neil (NYU), Iain Murray (Edinburgh), Malz, Vakili, and I met for coffee to talk cosmology and cosmological inference. We discussed the linear algebra of making full (non-approximate) likelihood calls for cosmic background inference, which includes determinant and solve calls for enormous (dense) matrices. A few days ago, Wandelt made the (perhaps obscure) comment to me that he did his Gibbs sampling in the CMB to avoid making determinant calls. I finally understood this: The determinant is big and expensive in part because it is like an integral or marginalization over the nuisance parameters (which are the initial conditions or phases). If you can compute the determinant, you get something you can think of as being a marginalized likelihood function. The Gibbs sampling is part of a system that does posterior sampling, so it does the marginalization but doesn't return the amplitude of the likelihood function. If you need a marginalized likelihood function, you can't do it with the Gibbs sampling. Murray once again made the simple point that the posterior (for the cosmological parameters) will in general be much easier to compute and represent than the likelihood function (the probability of the data given the cosmological parameters) because the former is a much smaller-dimensional thing. That's a deep point!

That deep point played also into his talk in the afternoon about his RNADE method (and code) to use deep learning to represent very complex densities, or estimate a density given a sampling. One of his application areas is a project to obtain posterior constraints on the mass of the Milky Way given (as data) the properties of the Magellanic Clouds. The theory in this case is represented by a large number of numerical simulations, but Murray wants a continuous density to do inference. RNADE is really impressive technology.