# Improved renormalization group computation of likelihood functions for cosmological data sets

## TL;DR

This paper enhances a renormalization group method for efficiently computing likelihood functions in large cosmological data sets by reducing computational complexity and improving accuracy, enabling rapid analysis of million-cell data.

## Contribution

The paper introduces a refined renormalization group approach that integrates out differences between specific adjacent cells, significantly improving computational efficiency and accuracy for large-scale likelihood evaluations.

## Key findings

- Likelihood computation for 1 million cells takes 2 minutes on a laptop.
- Method achieves linear scaling with data set size.
- Potential for further optimization and parallelization.

## Abstract

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.