Data analysis recipes: Fitting a model to data

TL;DR

This paper discusses advanced methods for fitting models to data, emphasizing the importance of generative models and likelihood-based approaches in complex, real-world scenarios with heterogeneous uncertainties and outliers.

Contribution

It provides a comprehensive overview of fitting techniques beyond standard least squares, including handling of complex uncertainties, outliers, and the use of generative models for robust inference.

Findings

Standard least-squares fitting is often inadequate for complex data.

Generative models enable more accurate likelihood-based fitting.

Handling of outliers and unknown uncertainties improves model reliability.

Abstract

We go through the many considerations involved in fitting a model to data, using as an example the fit of a straight line to a set of points in a two-dimensional plane. Standard weighted least-squares fitting is only appropriate when there is a dimension along which the data points have negligible uncertainties, and another along which all the uncertainties can be described by Gaussians of known variance; these conditions are rarely met in practice. We consider cases of general, heterogeneous, and arbitrarily covariant two-dimensional uncertainties, and situations in which there are bad data (large outliers), unknown uncertainties, and unknown but expected intrinsic scatter in the linear relationship being fit. Above all we emphasize the importance of having a "generative model" for the data, even an approximate one. Once there is a generative model, the subsequent fitting is non-arbitrary because the model permits direct computation of the likelihood of the parameters or the posterior probability distribution. Construction of a posterior probability distribution is indispensible if there are "nuisance parameters" to marginalize away.