# The exact non-Gaussian weak lensing likelihood: A framework to calculate analytic likelihoods for correlation functions on masked Gaussian random fields

arXiv: https://arxiv.org/abs/2407.08718 · HTML: https://mlsanity.com/paper/2407.08718/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2024.

## Built on by (most cited first, 1 of 1)

- [Dark Energy Survey Year 3 results: Simulation-based wCDM inference from weak lensing and galaxy clustering maps with deep learning: Analysis design](https://mlsanity.com/paper/2511.04681/) (2025)

## Abstract (arXiv)

We present exact non-Gaussian joint likelihoods for auto- and cross-correlation functions on arbitrarily masked spherical Gaussian random fields. Our considerations apply to spin-0 as well as spin-2 fields but are demonstrated here for the spin-2 weak-lensing correlation function. We motivate that this likelihood cannot be Gaussian and show how it can nevertheless be calculated exactly for any mask geometry and on a curved sky, as well as jointly for different angular-separation bins and redshift-bin combinations. Splitting our calculation into a large- and small-scale part, we apply a computationally efficient approximation for the small scales that does not alter the overall non-Gaussian likelihood shape. To compare our exact likelihoods to correlation-function sampling distributions, we simulated a large number of weak-lensing maps, including shape noise, and find excellent agreement for one-dimensional as well as two-dimensional distributions. Furthermore, we compare the exact likelihood to the widely employed Gaussian likelihood and find significant levels of skewness at angular separations gtrsim 1circ, such that the mode of the exact distributions is shifted away from the mean towards lower values of the correlation function. We find that the assumption of a Gaussian random field for the weak-lensing field is well valid at these angular separations. Considering the skewness of the non-Gaussian likelihood, we evaluate its impact on the posterior constraints on S8. On a simplified weak-lensing-survey setup with an area of 10  000  deg2, we find that the posterior mean of S8 is around 2.5% higher when using the non-Gaussian likelihood, a shift comparable to the precision of current stage-III surveys.