Satellite conjunction analysis and the false confidence theorem

Abstract

Satellite conjunction analysis is the assessment of collision risk during a close encounter between a satellite and another object in orbit. A counterintuitive phenomenon has emerged in the conjunction analysis literature: probability dilution, in which lower quality data paradoxically appear to reduce the risk of collision. We show that probability dilution is a symptom of a fundamental deficiency in epistemic probability distributions. In probabilistic representations of statistical inference, there are always false propositions that have a high probability of being assigned a high degree of belief. We call this deficiency false confidence. In satellite conjunction analysis, it results in a severe and persistent underestimation of collision risk exposure.We introduce the Martin-Liu validity criterion as a benchmark by which to identify statistical methods that are free from false confidence. If expressed using belief functions, such inferences will necessarily be non-additive. In satellite conjunction analysis, we show that Kσ uncertainty ellipsoids satisfy the validity criterion. Performing collision avoidance maneuvers based on ellipsoid overlap will ensure that collision risk is capped at the user-specified level. Further, this investigation into satellite conjunction analysis provides a template for recognizing and resolving false confidence issues as they occur in other problems of statistical inference.

False confidence, non-additive beliefs, and valid statistical inference

“…Before having put together the results presented in Balch et al (2017) and the valid inferential model construction in Martin and Liu (2016), I would've said that this "convergence of view" ship had sailed. The difference is that now we understand what can go wrong in practice with additivity, not just philosophies and generalities, along with a normative approach for the construction of valid inferential models.…”

“…The next step in the analysis, as reviewed in Balch et al (2017), is to first construct a corresponding posterior distribution for θ. Since θ is a location parameter, one might consider an inferential model that describes degrees of belief based on (θ | y) ∼ Π y := N 2 (y, Σ).…”

“…Theorem 1 (Balch et al 2017). Consider probability measure Π y with a density function that is bounded and continuous for each y.…”

An exposition of the false confidence theorem

“…In a recent paper, Balch et al () presents the phenomenon of “false confidence” associated with Bayesian posterior uncertainty. The authors come about the concept of false confidence from an alarming application to satellite collision risk analysis when estimating the posterior probability of the event that two satellites will collide.…”

“… A recent paper (Balch et al (), ‘Satellite conjunction analysis and the false confidence theorem’, arXiv 5 preprint arXiv:1706.08565.) presents the “false confidence theorem,” which has potentially broad implications for statistical inference using Bayesian posterior uncertainty.…”