This thesis studies statistical inference in the high energy physics unfolding problem, which is an ill-posed inverse problem arising in data analysis at the Large Hadron Collider (LHC) at CERN. Any measurement made at the LHC is smeared by the finite resolution of the particle detectors and the goal in unfolding is to use these smeared measurements to make nonparametric inferences about the underlying particle spectrum. Mathematically the problem consists in inferring the intensity function of an indirectly observed Poisson point process. Rigorous uncertainty quantification of the unfolded spectrum is of central importance to particle physicists. The problem is typically solved by first forming a regularized point estimator in the unfolded space and then using the variability of this estimator to form frequentist confidence intervals. Such confidence intervals, however, underestimate the uncertainty, since they neglect the bias that is used to regularize the problem. We demonstrate that, as a result, conventional statistical techniques as well as the methods that are presently used at the LHC yield confidence intervals which may suffer from severe undercoverage in realistic unfolding scenarios. We propose two complementary ways of addressing this issue. The first approach applies to situations where the unfolded spectrum is expected to be a smooth function and consists in using an iterative bias-correction technique for debiasing the unfolded point estimator obtained using a roughness penalty. We demonstrate that basing the uncertainties on the variability of the bias-corrected point estimator provides significantly improved coverage with only a modest increase in the length of the confidence intervals, even when the amount of bias-correction is chosen in a data-driven way. We compare the iterative bias-correction to an alternative debiasing technique based on undersmoothing and find that, in several situations, bias-correction provides shorter confidence intervals than undersmoothing. The new methodology is applied to unfolding the Z boson invariant mass spectrum measured in the CMS experiment at the LHC. The second approach exploits the fact that a significant portion of LHC particle spectra are known to have a steeply falling shape. A physically justified way of regularizing such spectra is to impose shape constraints in the form of positivity, monotonicity and convexity. Moreover, when the shape constraints are applied to an unfolded confidence set, one can regularize the length of the confidence intervals without sacrificing coverage. More specifically, we form shape-constrained confidence intervals by considering all those spectra that satisfy the shape constraints and fit the smeared data within a given confidence level. This enables us to derive regularized unfolded uncertainties which have by construction guaranteed simultaneous finite-sample coverage, provided that the true spectrum satisfies the shape constraints. The uncertainties are conservative, but still usefully tight. The method is demonstrated using simulations designed to mimic unfolding the inclusive jet transverse momentum spectrum at the LHC.