In decision-making situations characterized by uncertainty and risk, sensitivity analyses not only reveal which inputs most affect outcomes but also pinpoint where additional information could lead to better informed decisions. One powerful approach to evaluating the benefit of additional information is the Value of Information (VoI), which quantifies the expected enhancement in decision quality. While the VoI can significantly enhance data acquisition strategies, its major drawback remains its high computational complexity. VoI evaluation necessitates integration over potential observation outcomes for each candidate strategy, incorporating optimization over possible actions based on the expected costs for the action-measurement combinations. We focus on efficient estimation of the outermost integral over potential observation outcomes. Estimating this integral with brute-force Monte Carlo sampling is inefficient in cases where several measurements have more than two possible outcomes. We propose a novel importance sampling scheme that sequentially approximates the optimal importance sampling density using a Gaussian process surrogate model of the logarithmically transformed failure probability. Thereby, the sequentially drawn training points and their corresponding failure probability estimates serve as both importance samples and updates to the surrogate model. Using engineering test cases, we demonstrate the method’s potential to accelerate VoI calculations, with the hope that its implementation can enable more frequent usage in practice.