When knowing less looks safer

Here is a property of conjunction assessment that sounds like a bug and is not: for a given close approach, increasing your uncertainty about where the objects are can decrease the computed probability of collision.

Worse tracking, safer-looking number. This is called probability dilution, it is well documented, and it is the sharpest example I know of a metric that inverts under exactly the conditions where you most need it.

Where the number comes from

Collision probability for a short-duration encounter is computed by projecting both objects’ position uncertainties into the plane perpendicular to their relative velocity, combining them into a single covariance, and integrating that distribution over a circle representing the combined hard-body radius of the two objects.

In other words: how much probability mass falls inside a small disc.

Why more uncertainty can lower it

A probability density integrates to one. Spread it over a wider area and it is lower everywhere.

If the covariance is tight and centred near the miss distance, a large share of the mass sits inside the hard-body disc, and the computed probability is high. Inflate the covariance — because tracking is sparse, or the propagation is long, or the drag model is uncertain — and the same mass spreads across a much larger region. The fraction inside the disc falls.

So the number goes down. Not because the encounter got safer, but because you know less about it.

There is a genuinely perverse consequence: as covariance grows without bound, computed collision probability tends toward zero. Total ignorance produces a reassuring number.

The non-monotonic bit

The relationship is not simply decreasing. Starting from a very tight covariance and widening it, probability often rises first — as the distribution spreads to cover a miss distance it previously excluded — peaks, and only then falls as dilution takes over.

Which means a single probability value can correspond to two very different situations on opposite sides of that peak: one where you know a lot and the risk is genuinely low, and one where you know very little. The number alone does not distinguish them.

Operational practice handles this with the dilution region concept — identifying where on that curve an event sits, rather than reading the probability alone. Hejduk and colleagues at NASA have written the most accessible treatments.

The stronger critique

Balch, Martin and Ferson pushed this further in a 2019 Proceedings of the Royal Society A paper, arguing that the standard approach exhibits what they call false confidence: the probability can be made arbitrarily small by degrading the data, so a decision rule keyed to a probability threshold is systematically exploitable by poor measurement.

The paper prompted substantial disagreement, and I am not equipped to adjudicate it. What is not disputed is the underlying behaviour of the integral. The argument is about how much it matters in practice and what should replace the threshold rule, not about whether dilution happens.

Why I keep coming back to this

It is the cleanest real-world case of a pathology I care about generally: a summary statistic that is well defined, correctly computed, and misleading in exactly the regime where a decision depends on it.

The pattern is not specific to orbits. Any metric that normalises by a quantity related to uncertainty can do this. A calibrated model that reports 3% is making a different claim from an ignorant model that reports 3%, and if the interface shows one number, the reader cannot tell which they are holding.

It is also why I keep the two apart in my own work rather than folding confidence into the score it describes. That does not solve dilution — different problem, different pipeline — but it follows the same principle: never let a confidence estimate silently modulate the value it is attached to.

References

1. S. Alfano. “Review of Conjunction Probability Methods for Short-term Encounters.” Advances in the Astronautical Sciences, AAS 07-148, 2007. The standard survey of how the integral is computed.
2. F. R. Hoots, R. L. Roehrich. Spacetrack Report No. 3: Models for Propagation of NORAD Element Sets, 1980. Where the propagation uncertainty originates.
3. M. D. Hejduk et al. Work on conjunction assessment risk analysis and the dilution region, NASA Conjunction Assessment Risk Analysis programme. The most practical treatment of reading probability alongside covariance.
4. M. S. Balch, R. Martin, S. Ferson. “Satellite conjunction analysis and the false confidence theorem.” Proceedings of the Royal Society A, 475(2227), 2019. The formal critique, and worth reading alongside the responses to it.
5. J. L. Foster, H. S. Estes. A Parametric Analysis of Orbital Debris Collision Probability and Maneuver Rate for Space Vehicles. NASA JSC-25898, 1992. Origin of the threshold conventions still in use.