Study of constraint and impact of a nuisance parameter in maximum likelihood method

Maximum likelihood method is widely used for parameter estimation in high energy physics. To consider various systematic uncertainties, tens of or even hundreds of nuisance parameters (NP) are introduced in a likelihood fit. The constraint of a nuisance parameter and its impact on the parameter of interest (POI) will be the main concerns for a precise measurement. A fit involving many parameters is usually slow and it is even more time-consuming to investigate why a parameter is over-constrained or has a large impact. In this paper, we are trying to understand the reasons behind and provide simple formulae to estimate the constraint and impact directly.