Incorporating Nuisance Parameters in Likelihoods for Multisource Spectra

We describe here the general mathematical approach to constructing likelihoods for fitting observed spectra in one or more dimensions with multiple sources, including the effects of systematic uncertainties represented as nuisance parameters, when the likelihood is to be maximized with respect to these parameters. We consider three types of nuisance parameters: simple multiplicative factors, source spectra “morphing” parameters, and parameters representing statistical uncertainties in the predicted source spectra.

💡 Research Summary

The paper presents a comprehensive framework for constructing binned Poisson likelihoods to fit observed spectra that may be multidimensional and contain contributions from multiple sources. Starting from the basic likelihood L = ∏₁ᴺ P(n_i | μ_i), where each bin count n_i follows a Poisson distribution with expectation μ_i, the authors express μ_i as the product of integrated luminosity L, source cross‑section σ_j, and bin‑wise efficiency ε_{ji} obtained from Monte‑Carlo (MC) simulation. Systematic uncertainties are incorporated through nuisance parameters of three distinct types.

First, multiplicative uncertainties (e.g., a 2 % luminosity error) are modeled by promoting the physical quantity (such as L) to a free parameter constrained by a Gaussian prior G(L | L̃, σ_L). This adds a penalty term (L − L̃)²/(2σ_L²) to the negative log‑likelihood, effectively keeping the parameter near its measured value while allowing it to float during the fit. The same approach can be applied to cross‑sections, overall efficiencies, or any scale factor, with alternative priors (log‑normal, etc.) used when positivity must be enforced.

Second, shape‑changing systematics are handled via a “morphing” parameter f. By generating MC samples with the systematic shifted by ±1 σ (yielding efficiencies ε_{−}, ε_{0}, ε_{+}), the efficiency in each bin is interpolated as a continuous function of f. The authors adopt a Lagrange interpolation that is quadratic for |f| < 1 and linear beyond, ensuring the exact values at f = ±1 while avoiding large extrapolation errors. Multiple morphing parameters can be added linearly to model several independent effects.

Third, statistical uncertainties in the MC‑derived efficiencies are treated using the Barlow‑Beeston method. Each source‑bin receives a scaling factor β_{ji} with a Gaussian (or Poisson) constraint centered at 1. The negative log‑likelihood then contains a term (β_{ji} − 1)²/(2σ_{ji}²). Solving for the β_{ji} that maximize the likelihood leads to a set of nonlinear equations per bin, which can be tackled with Newton‑type iterations. In practice, however, MINUIT’s MIGRAD algorithm can encounter discontinuities in β, causing the Hessian to become non‑positive‑definite and the fit to diverge. To mitigate this, the authors propose collapsing the many β parameters into a single overall statistical nuisance β per bin, representing the combined uncertainty of all sources. This reduces the problem to solving a quadratic equation analytically, preserving stability while still accounting for MC statistical errors.

The paper also discusses practical pitfalls. Sparse or empty bins can cause morphing‑induced migrations that break the likelihood; the authors recommend either generating sufficient MC statistics or merging bins according to a pre‑defined algorithm that does not use the observed data. They stress the importance of fixing the set of bins used in the fit beforehand, ensuring that no bin with observed data lacks any expected contribution, and suggest imposing tiny floor values (e.g., 10⁻¹⁰ events) to avoid log‑zero problems.

Finally, the authors address the interpretation of the profile likelihood as a pseudo‑Bayesian posterior. By treating the maximized likelihood L_max(σ_X) as a function of a parameter of interest (e.g., a new particle’s cross‑section) and multiplying by a prior, one obtains a posterior density that closely matches the result of a full marginalization over nuisance parameters, but at a fraction of the computational cost. Empirical tests on complex spectrum fits confirm the near‑identity of the two approaches.

In summary, the work provides a mathematically rigorous yet computationally tractable method for incorporating multiplicative, shape‑changing, and statistical systematic uncertainties into multi‑source spectral fits, outlines implementation details within MINUIT, and offers practical guidance for handling sparse data and ensuring robust convergence.