Reasoning about Uncertainty in Metric Spaces
This work addresses foundational issues in AI and ML for applications requiring uncertainty reasoning in structured spaces, but it appears incremental as it builds on existing metric and belief theory frameworks.
The paper tackles the problem of reasoning about uncertainty in metric spaces by integrating belief theoretic measures with both probabilistic and metric aspects, resulting in a formal logical system that is proven sound and complete, and a new metric for product spaces with good properties.
We set up a model for reasoning about metric spaces with belief theoretic measures. The uncertainty in these spaces stems from both probability and metric. To represent both aspect of uncertainty, we choose an expected distance function as a measure of uncertainty. A formal logical system is constructed for the reasoning about expected distance. Soundness and completeness are shown for this logic. For reasoning on product metric space with uncertainty, a new metric is defined and shown to have good properties.