On the combination of correlated estimates of a physics observable

TL;DR

The paper analyzes how Best Linear Unbiased Estimation (BLUE) combines correlated estimates of a single observable, focusing on the two-estimator case to reveal how the weight of the less precise estimate, $eta$, and the uncertainty ratio $rac{\sigma_x}{\sigma_1}$ depend on the correlation $ ho$ and the relative precision $z$. It derives explicit formulas for $eta$ and $rac{\sigma_x}{\sigma_1}$, discusses the role of conditional probabilities in Peelle's Pertinent Puzzle, and assesses absolute versus relative uncertainty modeling. It critically evaluates proposals like reduced correlations and variance-maximizing techniques, arguing they can be unphysical or overly conservative, and offers a practical procedure to decide which estimates to combine, including per-source stability checks and a TEV-1301 top-quark mass example. The authors provide a software tool to perform these analyses and advocate a per-source, stability-driven approach to combining correlated estimates in physics analyses. The work informs best practices for combining results across experiments and observables, with implications for precision measurements in high-energy physics.

Abstract

The combination of a number of correlated estimates of a given observable is frequently performed using the Best Linear Unbiased Estimate (BLUE) method. Most features of such a combination can already be seen by analysing the special case of a pair of estimates from two correlated estimators of the observable. Two important parameters of this combination are the weight of the less precise estimate and the ratio of uncertainties of the combined result and the more precise estimate. Derivatives of these quantities are derived with respect to the correlation and the ratio of uncertainties of the two estimates. The impact of using either absolute or relative uncertainties in the BLUE combination is investigated on a number of examples including Peelle's Pertinent Puzzle. Using an example, a critical assessment is performed of suggested methods to deal with the fact that both the correlation and the ratio of uncertainties of a pair of estimates are typically only known with some uncertainty. Finally, a proposal is made to decide on the usefulness of a combination and to perform it. The proposal is based on possible improvements with respect to the most precise estimate by including additional estimates. This procedure can be applied to the general case of several observables.

Figures (7)

• Figure 1: The two-dimensional pdf ${\@fontswitch\mathcal{P}}(X_\mathrm{1}\xspace, X_\mathrm{2}\xspace)$ for three values of the correlation $\rho\xspace$ obtained using five million pairs of estimates. The black line corresponds to $X_\mathrm{1}\xspace=X_\mathrm{2}\xspace$, the red line to $X_\mathrm{1}\xspace=x_\mathrm{1}\xspace$, and finally the dot to a particular pair of estimates chosen to be $x_\mathrm{1}\xspace=0.30$ and $x_\mathrm{2}\xspace=0.95$. The variable $f_\mathrm{out}$ denotes the fraction of events for which $x_\mathrm{T}$ does not lie within the interval spanned by the pair of estimates. Shown are (a) $\rho\xspace=0$, (b) $\rho\xspace=0.9$, and (c) $\rho\xspace=-0.9$. In b (c) the half axes shown in blue are changed and rotated (counter) clockwise from the positive $X_\mathrm{2}$ axis.
• Figure 2: The results for Eqs. \ref{['eq:beta']}, \ref{['eq:sigx']}--\ref{['eq:dsdr']} as functions of $\rho$ for a number of $z$ values. Shown are (a) $\beta$ and (b) $\sigma_{x}$/$\sigma_{1}$ and their derivatives with respect to $\rho$, (c) $\partial\beta\xspace/\partial\rho\xspace$ and (d) $1/\sigma_{1}\xspace\ \partial\sigma_{x}\xspace/\partial\rho\xspace$.
• Figure 3: The results for Eqs. \ref{['eq:beta']}, \ref{['eq:sigx']}, \ref{['eq:dbdz']}--\ref{['eq:dsdz']} as functions of $z$ for a number of $\rho$ values. Shown are (a) $\beta$ and (b) $\sigma_{x}$/$\sigma_{1}$ and their derivatives with respect to $z$, (c) $\partial\beta\xspace/\partial z$ and (d) $1/\sigma_{1}\xspace\ \partial\sigma_{x}\xspace/\partial z$.
• Figure 4: Results of Peelle's Pertinent Puzzle for scenario $\@fontswitch\mathcal{A}$ for one hundred thousand pairs of estimates. The simulation is based on a hypothetical two-dimensional pdf assuming $x_\mathrm{T}\xspace=1$, using the uncertainties and correlation of the estimates from this scenario, and simulating absolute uncertainties. Shown are (a) the two-dimensional distribution of the pairs of estimates, (b) the $\chi^2\xspace(X_\mathrm{1}\xspace, X_\mathrm{2}\xspace)$ of the pairs of estimates, (c) the two-dimensional distribution of the pairs of combined results when using either absolute uncertainties (X), or relative uncertainties (Y), and (d) the $\chi^2\xspace(X, Y)$ of the pairs of results. Both $\chi^2$ distributions are truncated at $\chi^2\xspace=8$. The red points correspond to the estimates (a) and combined results (c) for this scenario, see Table \ref{['tab:BluePeel']}. In addition listed for the estimates are in (a) their mean values and uncertainties together with their correlation, and in (b) the fraction of pairs for which the $\chi^2$ value exceeds the one observed for this scenario. The analogous quantities for the combined results are given in (c) and (d), respectively.
• Figure 5: Results for the Blue combination using the hypothetical example from Table \ref{['tab:BlueRes']}, scenario $\@fontswitch\mathcal{A}$. The sub-figures (a)--(h) correspond to Figures \ref{['fig:bsvsr']}--\ref{['fig:bsvsz']} for the pair of estimates investigated. The black points represent the actual values of the parameter shown at the given values of $\rho$ and $z$. In (a) also the estimates $x_\mathrm{1}$ and $x_\mathrm{2}$, as well as the combined value $x$, together with their uncertainties, are listed. In each sub-figure three curves are shown in which, for parameters shown as a function of $\rho$ (or $z$), the value of $z$ (or $\rho$) is varied. The curves corresponding to the minimum/central/maximum value of this variation are shown in blue/black/red, and the three values used for $z$ and $\rho$ are given in (b) and (d), respectively. For the derivatives of $\beta$ and $\sigma_{x}$/$\sigma_{1}$ with respect to $\rho$ and $z$, for each sub-figure the range of observed parameter values is given. This range is obtained for the three curves shown, while keeping the respective value of the other parameter. As an example in (b) the range in $\partial\,\beta\xspace/\partial\,\rho$ at $\rho\xspace=0.78$ is quoted observed when changing $z$ from 1.39 to 1.69. Finally, for $\beta$ and $\sigma_{x}$/$\sigma_{1}$ their full range is quoted in (a) and (e). This range is obtained using all nine possible pairs of the $\rho$ and $z$ values.