Uncertainty quantification for wide-bin unfolding: one-at-a-time strict bounds and prior-optimized confidence intervals

Abstract

Unfolding is an ill-posed inverse problem in particle physics aiming to infer a true particle-level spectrum from smeared detector-level data. For computational and practical reasons, these spaces are typically discretized using histograms, and the smearing is modeled through a response matrix corresponding to a discretized smearing kernel of the particle detector. This response matrix depends on the unknown shape of the true spectrum, leading to a fundamental systematic uncertainty in the unfolding problem. To handle the ill-posed nature of the problem, common approaches regularize the problem either directly via methods such as Tikhonov regularization, or implicitly by using wide-bins in the true space that match the resolution of the detector. Unfortunately, both of these methods lead to a non-trivial bias in the unfolded estimator, thereby hampering frequentist coverage guarantees for confidence intervals constructed from these methods. We propose two new approaches to addressing the bias in the wide-bin setting through methods called One-at-a-time Strict Bounds (OSB) and Prior-Optimized (PO) intervals. The OSB intervals are a bin-wise modification of an existing guaranteed-coverage procedure, while the PO intervals are based on a decision-theoretic view of the problem. Importantly, both approaches provide well-calibrated frequentist confidence intervals even in constrained and rank-deficient settings. These methods are built upon a more general answer to the wide-bin bias problem, involving unfolding with fine bins first, followed by constructing confidence intervals for linear functionals of the fine-bin counts. We test and compare these methods to other available methodologies in a wide-bin deconvolution example and a realistic particle physics simulation of unfolding a steeply falling particle spectrum.

Statistical constraints on climate model parameters using a scalable cloud-based inference framework

“…Specifically, our work deals with a special case of simulator-based inference where the observations are given by a deterministic simulator and an additive noise model. Patil et al (2022) and Stanley et al (2022) use a strict bounds method (Stark, 1992) to construct efficient confidence sets for the model parameters in closely related inverse problems in remote sensing and high energy physics. Unlike in these works where the forward models of interest are linear and known exactly, the present problem features a forward model (UKESM1) which is nonlinear and estimated using an emulator.…”

Post-hoc regularisation of unfolded cross-section measurements

“…Depending on how this is done, it could also lead to increased model dependencies if the detector response varies considerably within a single analysis bin. The latter can be avoided by unfolding with fine bins and then combining them into larger bins for presentation [4]. In that sense, the re-binning is a form of regularisation, where joined cross-section bins are forced to have identical values.…”