In this thesis, models for uncertainty quantification in the case of scarce and imprecise data are described, and the computational efficiency of simulations with these models is improved. Specifically, probability boxes are used to describe imprecision in cumulative distribution functions. This may be the case when imprecise data is used to train a model, or the prior knowledge regarding a property of the system being studied is very weak. Performing simulations with probability boxes is often computationally expensive, because an optimisation program must be solved to obtain each sample in a Monte Carlo simulation. When the system model is known analytically, it is possible to significantly reduce the cost of the analysis. However, the system model is often a black box which can only be queried for a particular point value of the input. Each evaluation or query of the system model is often computationally expensive in itself. Currently, few efficient methods exist to perform computations with probability boxes, and the techniques which exist do not provide rigorous bounds on the obtained probability of failure. Interval Predictor Models are a technique to create an approximate representation of a function, where the uncertainty in the true function is described as an interval, with statistical guarantees on the coverage of the true function. This thesis proposes the use of Interval Predictor Models to create an approximate surrogate model for the true black box system model and hence obtain rigorous bounds on the probability of failure of a system. Techniques are described to create Interval Predictor Models which are tailored to model the performance of a system for reliability analysis. This thesis also describes analytical techniques which can be used for probabilistic safety analysis, in the case that the system model is not a black box. This is advantageous as it enables engineers to perform calculations without spending time programming complex Monte Carlo simulations. A technique is presented to efficiently create Interval Predictor Models for datasets of arbitrary complexity and size, which may contain imprecise data, and we call these models Interval Neural Networks. Case studies or numerical examples are presented to demonstrate the performance of the proposed techniques, including some common benchmarks and a finite element model. The interval predictor models used in this thesis were implemented in the open source uncertainty quantification software OpenCossan, and are now freely available.