Decisions about health interventions are often made using limited evidence. Mathematical models used to inform such decisions often include uncertainty analysis, i.e., probabilistic sensitivity analysis (PSA), to evaluate the effect of uncertainty in the current evidence base on decisional-relevant quantities. However, PSA requires modelers to specify a precise probability distribution to represent the uncertainty of a model parameter. This requirement may not be amenable to situations where data is limited or does not exist. This study introduces a novel approach for propagating parameter uncertainty, probability bounds analysis (PBA), where the uncertainty about the unknown probability distribution of a model parameter is expressed in terms of an interval that is bounded by lower and upper bounding functions on the unknown cumulative distribution function (CDF) and without assuming a particular form of the CDF. We give the formulas for calculating the bounds on the unknown CDF for common data availability situations (given combinations of data on minimum, maximum, median, mean, or standard deviation), describe an approach to propagate the bounds into a black-box mathematical model, and introduce one approach for decision-making using the results of PBA. Then, we demonstrate an application of PBA using a case study where we compare PBA vs. PSA. We demonstrate that using PBA results in bounds on the unknown CDF of the quantity of interest (QoI). In contrast, the commonly used PSA produces a tighter uncertainty around the QoI, which assumes more information (the functional form of the CDF) than that is actually available. PBA also provides additional information: the tightest interval that contains the true QoI, given some minimal data on the model parameters. In sum, this study provides practitioners in health economic evaluation and decision analysis with a novel method to conduct parameter uncertainty quantification given constraints of available data and with the fewest assumptions.