The assessments for the various conditional probabilities of a Bayesian belief network inevitably are inaccurate, influencing the reliability of its output. By subjecting the network to a sensitivity analysis with respect to its conditional probabilities, the reliability of its output can be investigated. Unfortunately, straightforward sensitivity analysis of a belief network is highly time-consuming. In this paper, we show that by qualitative considerations several analyses can be identified as being uninformative as the conditional probabilities under study cannot affect the output. In addition, we show that the analyses that are informative comply with simple mathematical functions. More specifically, we show that a belief network's output can be expressed as a quotient of two functions that are linear in a conditional probability under study. These properties allow for considerably reducing the computational burden of sensitivity analysis of Bayesian belief networks.