HDSense: An efficient method for ranking observable sensitivity

TL;DR

This work tackles the problem of identifying the most constraining observables for model parameters when full joint likelihoods with correlations are intractable. It introduces HDSense, a score that uses only marginal one-dimensional histograms by approximating the Fisher information: ${\cal S}_{\rm HD}({\cal X}) = \mathrm{Info}({\cal X}) [1-\beta \mathcal{P}_{\mathrm{overlap}}({\cal X})]$, with ${\rm Info}({\cal X}) = \sum_{i \in \mathcal X} \mathrm{Tr}[I^{(i)}]$ and ${\cal P}_{\mathrm{overlap}}({\cal X}) = \frac{2}{\sum_{k \in \mathcal{X}} \mathrm{Tr}[I^{(k)}]} \sum_{i<j} \sqrt{\mathrm{Tr}[I^{(i)}] \mathrm{Tr}[I^{(j)}] \cos(\Phi^F_{ij})}$ and $\cos(\Phi^F_{ij}) = \frac{\langle I^{(i)}, I^{(j)}\rangle_F}{||I^{(i)}||_F ||I^{(j)}||_F}$. The penalty strength is set by $\beta = \beta_0 / \max_{\mathcal X}{\cal P}_{\mathrm{overlap}}({\mathcal X})$ with $\beta_0 = 0.5$, and is adapted with the number of observables to reflect correlation importance. The method is validated against approximate full-likelihood approaches in a Lund string hadronization setting (Pythia) at the $Z$ pole, demonstrating near-optimal observable subset identification and the ability to combine measurements across experiments and account for detector effects. Although model-dependent, HDSense provides a robust, scalable tool for parameter inference problems with unknown correlations and has potential applications beyond hadronization, including astrophysical and cosmological modeling.

Abstract

Identifying which observables most effectively constrain model parameters can be computationally prohibitive when considering full likelihoods of many correlated observables. This is especially important for, e.g., hadronization models, where high precision is required to interpret the results of collider experiments. We introduce the High-Dimensional Sensitivity (HDSense) score, a computationally efficient metric for ranking observable sets using only one-dimensional histograms. Derived by profiling over unknown correlations in the Fisher information framework, the score balances total information content against redundancy between observables. We apply HDSense to rank a set observables in terms of their constraining power with respect to five parameters of the Lund string model of hadronization implemented in Pythia using simulated leptonic collider events at the $Z$ pole. Validation against machine-learning--based full-likelihood approximations demonstrates that HDSense successfully identifies near-optimal observable subsets. The framework naturally handles data from multiple experiments with different acceptances and incorporates detector effects. While demonstrated on hadronization models, the methodology applies broadly to generic parameter estimation problems where correlations are unknown or difficult to model.

Figures (9)

• Figure 1: (Left) Covariance matrix $\Sigma$ corresponding to the toy example outlined in \ref{['eq:multivar_gaussian']}, with $K = 20$. The covariance is in block-diagonal form. (Right) The dependence of the means $\mu_i$ of each observable on each of the 5 parameters $\theta_i$.
• Figure 2: Validation of HDSense selection prescription against approximate full likelihood for $K = 3$ (top), $K = 5$ (middle), and $K = 7$ (bottom) observables. Left: Dependence of the full Fisher information trace on $\beta$ for the subset selected by HDSense at each $\beta$ value. The smallest and largest possible traces, obtained by exploring all possible $K$-observable subsets, are shown for reference. Right: Trace versus determinant of $\hat{I}_{\text{full}}^{-1}$ for all possible $K$-observable subsets. Better subsets lie closer to the origin. Red cross shows the heuristic choice $\beta = 0.5/\max\mathcal{P}_{\text{overlap}}$.
• Figure 3: Comparison of the selected subsets for all $K$. For $K\leq 7$, the HDSense score hovers close to the optimal region, while it remains far from the 'worst' possible score for all $K$. The quality of the approximation degrades for large $K$ where correlations become more important. Here, the $\Delta$ metric is defined as the ratio between the difference in log space of the worst combination and the combination selected by HDSense and the difference in log space between the best and worst combinations. All information matrices are computed using the approximate full likelihoods.
• Figure 4: HDSense ranking for all 15 observables, as assessed across bootstrapped samples (see text). Top:HDSense score (blue) and heuristic $\beta$ (red) versus number of selected observables $K$. The maximum HDSense score is highlighted with an additional blue circle and a dashed vertical line is drawn at the corresponding $K$-value. Bottom: Heatmap showing the fractional score reduction $\Delta\mathcal{S}_{\text{HD}}/\mathcal{S}_{\text{HD}}$ when each observable is excluded from the selected subset. Gray cells denote observables not yet selected at that $K$. Observables are ordered by their entry into the greedy selection algorithm. A dashed vertical line is drawn at the $K$-value corresponding to the maximum HDSense score.
• Figure 5: HDSense ranking using only the 9 event-level observables. Layout and interpretation as in \ref{['fig:all_observables_no_detector']}.