Improved renormalization group computation of likelihood functions for
  cosmological data sets
  Evaluation of likelihood functions for cosmological large scale structure
data sets (including CMB, galaxy redshift surveys, etc.) naturally involves
marginalization, i.e., integration, over an unknown underlying random signal
field. Recently, I showed how a renormalization group method can be used to
carry out this integration efficiently by first integrating out the smallest
scale structure, i.e., localized structure on the scale of differences between
nearby data cells, then combining adjacent cells in a coarse graining step,
then repeating this process over and over until all scales have been
integrated. Here I extend the formulation in several ways in order to reduce
the prefactor on the method's linear scaling with data set size. The key
improvement is showing how to integrate out the difference between specific
adjacent cells before summing them in the coarse graining step, compared to the
original formulation in which small-scale fluctuations were integrated more
generally. I suggest some other improvements in details of the scheme,
including showing how to perform the integration around a maximum likelihood
estimate for the underlying random field. In the end, an accurate likelihood
computation for a million-cell Gaussian test data set runs in two minutes on my
laptop, with room for further optimization and straightforward parallelization.
