On a unified framework for linear nuisance parameters

TL;DR

This paper presents a unified analytical framework for three common methods of handling linear nuisance parameters in estimation problems, revealing their equivalence and challenging some traditional assumptions.

Contribution

It introduces a general differential approach, analyzes the relations between methods, and shows their equivalence after whitening, with implications for various applications.

Findings

All three methods are equivalent after whitening.

The choice of reference in differencing is not crucial.

Localization examples confirm theoretical results.

Abstract

Estimation problems in the presence of deterministic linear nuisance parameters arise in a variety of fields. To cope with those, three common methods are widely considered: (1) jointly estimating the parameters of interest and the nuisance parameters, (2) projecting out the nuisance parameters, (3) selecting a reference and then taking differences between the reference and the observations, which we will refer to as "differential signal processing." A lot of literature has been devoted to these methods, yet all follow separate paths. Based on a unified framework, we analytically explore the relations between these three methods, where we particularly focus on the third one and introduce a general differential approach to cope with multiple distinct nuisance parameters. After a proper whitening procedure, the corresponding best linear unbiased estimators (BLUEs) are shown to be all equivalent to each other. Accordingly, we unveil some surprising facts, which are in contrast to what is commonly considered in literature, e.g., the reference choice is actually not important for the differencing process. Since this paper formulates the problem in a general manner, one may specialize our conclusions to any particular application. Some localization examples are also presented in this paper to verify our conclusions.