Estimating a bivariate linear relationship

TL;DR

This paper develops a Bayesian approach for estimating the slope in bivariate linear errors-in-variables models with unspecified errors, ensuring invariance under coordinate transformations and providing practical computation tools.

Contribution

It introduces a prior that respects invariance properties and derives a posterior for the slope based on sample correlation and standard deviation ratio, with an R package implementation.

Findings

Posterior density depends only on correlation and standard deviation ratio.

The posterior interval covers the two OLS estimates and diminishes outside.

Application to astronomy and method comparison demonstrates practical utility.

Abstract

Solutions of the bivariate, linear errors-in-variables estimation problem with unspecified errors are expected to be invariant under interchange and scaling of the coordinates. The appealing model of normally distributed true values and errors is unidentified without additional information. I propose a prior density that incorporates the fact that the slope and variance parameters together determine the covariance matrix of the unobserved true values but is otherwise diffuse. The marginal posterior density of the slope is invariant to interchange and scaling of the coordinates and depends on the data only through the sample correlation coefficient and ratio of standard deviations. It covers the interval between the two ordinary least squares estimates but diminishes rapidly outside of it. I introduce the R package leiv for computing the posterior density, and I apply it to examples in astronomy and method comparison.