Global Sensitivity Analysis of Uncertain Parameters in Bayesian Networks

The paper introduces a global sensitivity analysis method for Bayesian networks that overcomes the limitations of traditional one-at-a-time sensitivity analysis by using Sobol indices and low-rank tensor decomposition to efficiently handle uncertainties in multiple parameters simultaneously, thereby providing a more comprehensive understanding of parameter interactions and their impact on network outputs.

Abstract

Traditionally, the sensitivity analysis of a Bayesian network studies the impact of individually modifying the entries of its conditional probability tables in a one-at-a-time (OAT) fashion. However, this approach fails to give a comprehensive account of each inputs' relevance, since simultaneous perturbations in two or more parameters often entail higher-order effects that cannot be captured by an OAT analysis. We propose to conduct global variance-based sensitivity analysis instead, whereby n parameters are viewed as uncertain at once and their importance is assessed jointly. Our method works by encoding the uncertainties as n additional variables of the network. To prevent the curse of dimensionality while adding these dimensions, we use low-rank tensor decomposition to break down the new potentials into smaller factors. Last, we apply the method of Sobol to the resulting network to obtain n global sensitivity indices. Using a benchmark array of both expert-elicited and learned Bayesian networks, we demonstrate that the Sobol indices can significantly differ from the OAT indices, thus revealing the true influence of uncertain parameters and their interactions.

Method

The authors used a global variance-based sensitivity analysis methodology, specifically employing Sobol indices to assess the importance of uncertain parameters in Bayesian networks (BNs). They encoded these uncertainties as additional variables within the BN and utilized low-rank tensor decomposition techniques to manage the increased dimensionality and computational complexity that arises from adding these dimensions. This approach allowed them to evaluate the joint impact of multiple uncertain parameters and their interactions on the network's outputs.

Main Finding

The authors discovered that the Sobol indices, which are a measure of global sensitivity, can significantly differ from the traditional one-at-a-time (OAT) sensitivity indices. This discrepancy revealed the true influence of uncertain parameters and their interactions within Bayesian networks (BNs), which were not adequately captured by the OAT analysis. By applying their method to a benchmark array of both expert-elicited and learned BNs, they demonstrated that the proposed approach provides a more comprehensive and accurate assessment of parameter sensitivity, particularly in the context of complex models where multiple uncertainties are involved.

Conclusion

The conclusion of the paper by Rafael Ballester-Ripoll and Manuele Leonelli is that their proposed method for global sensitivity analysis in Bayesian networks, which utilizes Sobol indices and low-rank tensor decomposition, provides a significant advancement in the understanding of the influence of uncertain parameters and their interactions. The method allows for a more informed sensitivity analysis that accounts for the joint effects of multiple parameters, offering a valuable tool for applied analyses in various scientific fields. The authors also highlight the limitations of their approach, such as its inability to model uncertainties for multiple child state probabilities given the same configuration of parents, and suggest future research directions, including the exploration of approximate contraction schemes for larger networks and the computation of higher-order Sobol indices to discern important interactions between parameters.

Keywords

Bayesian networks, sensitivity analysis, Sobol indices, tensor networks, uncertainty quantification