# A geometric approach to non-linear correlations with intrinsic scatter

## TL;DR

This paper introduces a Riemannian geometric model for non-linear correlations with intrinsic scatter, enabling more accurate Bayesian estimation and application to astrophysical data, notably revising the slope of the black hole mass-velocity dispersion relation.

## Contribution

It presents a novel geometric framework for modeling non-linear correlations with intrinsic scatter, incorporating Bayesian methods and analytic likelihoods, applicable to complex datasets.

## Key findings

- The model is symmetric and invariant under coordinate transformations.
- It effectively incorporates censored and truncated data with measurement errors.
- Applied to astrophysical data, it suggests a higher slope (~6) for the black hole mass-velocity dispersion relation.

## Abstract

We propose a new mathematical model for $n-k$-dimensional non-linear correlations with intrinsic scatter in $n$-dimensional data. The model is based on Riemannian geometry, and is naturally symmetric with respect to the measured variables and invariant under coordinate transformations. We combine the model with a Bayesian approach for estimating the parameters of the correlation relation and the intrinsic scatter. A side benefit of the approach is that censored and truncated datasets and independent, arbitrary measurement errors can be incorporated. We also derive analytic likelihoods for the typical astrophysical use case of linear relations in $n$-dimensional Euclidean space. We pay particular attention to the case of linear regression in two dimensions, and compare our results to existing methods. Finally, we apply our methodology to the well-known $M_\text{BH}$-$\sigma$ correlation between the mass of a supermassive black hole in the centre of a galactic bulge and the corresponding bulge velocity dispersion. The main result of our analysis is that the most likely slope of this correlation is $\sim 6$ for the datasets used, rather than the values in the range $\sim 4$-$5$ typically quoted in the literature for these data.