Confidence in confidence distributions!

Abstract

The recent article `Satellite conjunction analysis and the false confidence theorem' (Balch, Martin, and Ferson, 2019, Proceedings of the Royal Society, Series A) points to certain difficulties with Bayesian analysis when used for models for satellite conjuntion and ensuing operative decisions. Here we supplement these previous analyses and findings with further insights, uncovering what we perceive of as being the crucial points, explained in a prototype setup where exact analysis is attainable. We also show that a different and frequentist method, involving confidence distributions, is free of the false confidence syndrome.

Figures (2)

• Figure 1: For each value of $\sigma$, we have computed the distribution of non-collision probabilities $1-B(2.00\,|\, y_1,y_2)$ and $1-C(2.00\,|\, y_1,y_2)$, with the Bayesian (slanted, red curves) and frequentist CD (full, black curves) methods. This is for a setup with $R=2.00$ the threshold for collision and true value $\delta=1.99$. Left panel: the means of these probabilities; right panel: the frequency of high probabilities, those above 0.95.
• Figure 2: Left panel: the CD (black curve) and the Bayesian posterior cumulative for $\delta$, after having observed $\|y\|=5.00$, with assumed $\sigma=2.50$; the critical value is $R=2.00$ (marked blue), where we read off the confidence 0.222 in $[0,R]$. Right panel: the corresponding confidence curve ${\rm cc}(\delta\,|\, y_1,y_2)=|1-2\,C(\delta\,|\, y_1,y_2)|$ (black) and the Bayesian credibility curve $|1-2\,B(\delta\,|\, y_1,y_2)|$ (red). Reading off 90% confidence and credibility intervals yields $[0.00,8.63]$ and $[2.01,9.57]$, respectively. The true $\delta=1.99$ behind the generation of $\|y\|$ here is indicated by the vertical blue line.