Elicitation of Probabilities for Belief Networks: Combining Qualitative and Quantitative Information (1302.4943v1)

Abstract: Although the usefulness of belief networks for reasoning under uncertainty is widely accepted, obtaining numerical probabilities that they require is still perceived a major obstacle. Often not enough statistical data is available to allow for reliable probability estimation. Available information may not be directly amenable for encoding in the network. Finally, domain experts may be reluctant to provide numerical probabilities. In this paper, we propose a method for elicitation of probabilities from a domain expert that is non-invasive and accommodates whatever probabilistic information the expert is willing to state. We express all available information, whether qualitative or quantitative in nature, in a canonical form consisting of (in) equalities expressing constraints on the hyperspace of possible joint probability distributions. We then use this canonical form to derive second-order probability distributions over the desired probabilities.

• The paper proposes a canonical form that converts diverse probabilistic inputs into constraints on a hyperspace of possible distributions.
• It demonstrates how to integrate qualitative expert judgments with quantitative data to estimate second-order probability distributions.
• Application to an HIV infection belief network illustrates enhanced flexibility and practical advantages in managing uncertain information.

Elicitation of Probabilities for Belief Networks: Combining Qualitative and Quantitative Information

The paper authored by Marek J. Druzdzel and Linda C. van der Gaag addresses a persistent challenge in artificial intelligence—eliciting probabilities for belief networks from domain experts. It seeks to provide a method that effectively integrates both qualitative and quantitative information to determine the probability distributions necessary for belief networks. This is a significant issue due to the insufficiency of statistical data available for accurate probability estimation, and experts' reluctance to provide precise numerical values.

Core Contributions

Druzdzel and van der Gaag propose a canonical form for probabilistic information representation, combining constraints on the hyperspace of possible joint probability distributions expressed as (in)equalities. The approach aims to reconcile a variety of probabilistic inputs, accommodating non-numerical and semi-numerical data as well as qualitative assertions, such as signs of influences and variable independence, into a cohesive probability elicitation process. The authors suggest that all available probabilistic input, regardless of its nature, can essentially be represented as constraints over a hyperspace, facilitating the derivation of second-order probability distributions.

Methodology and Application

The methodology is elucidated through an example belief network modeling HIV infection causes, encompassing variables like HIV infection, needle sharing, sexual intercourse, and condom usage. The development of a belief network typically involves constructing its qualitative (graphical representation of dependencies) and quantitative (probability distributions) components. The quantitative aspect poses more considerable challenges and is where this paper's contributions are most impactful.

The canonical form allows probabilistic information, such as conditional probabilities and qualitative relations, to be structured into a hyperspace of possible distributions. These constraints are then used to ascertain second-order probability distributions, addressing uncertainty in each probability of interest and providing an expected value for further decision-making processes.

Implications

Despite being computationally expensive, especially in cases of sparse or conflicting information, the proposed approach offers a flexible way to incorporate diverse forms of information into belief networks. By treating incomplete information as probabilistic constraints, it opens avenues for more nuanced and robust probability assessments. The paper highlights the conceptual and practical advantages of accommodating an expert's broad range of probabilistic statements, which may be presented incrementally as the elicitation process focuses more narrowly through iterative refinement.

This endeavor enhances the usability of belief networks by making the probabilistic quantification process less invasive and potentially more aligned with an expert's natural reasoning process. Such a method could lead to significant time and resource savings, given that traditional approaches often require extensive iterative consultations with domain experts to achieve consistent numerical probabilities.

Future Directions

The authors envision integrating the proposed elicitation mechanism into a computational tool for probabilistic reasoning in decision support systems. Future research directions could involve enhancing the efficiency of sampling algorithms used for deriving second-order distributions and exploring methods to handle inconsistencies and conflicts in expert-derived constraints. Improvements in computational techniques could curtail the otherwise high costs of this probabilistic reasoning approach.

Overall, this paper contributes a valuable methodological tool to the field of AI uncertainty modeling, facilitating more effective and flexible belief network quantification that leverages the best available probabilistic insights, regardless of their form.