[gyrovagueGeist]

To simplfy, let's assume you have perfect knowledge of everything else & that the only variable that matters is asteroids current position. By triangulating observations you have a point estimate. Due to calibrating your instruments in the past you know that they tend to have uniform additive noise that is the same in each dimension. Let's say it shifts measurements by up to 1km randomly.

So the best guess you have is that the true asteroid is 99% likely to be somewhere within a 2km box centered at the observation point.

For each possible location in this box you use it as a hypothetical starting point and run a simulation forward creating a trajectory. In 3% of these trajectories the asteroid hits the earth.

The 3% is only a probability over the measurement uncertainty. It represents our knowledge about the system in a bayesian sense. The true asteroid was always ever going to hit the earth or not. There is no uncertainty inherent in the system.

That many asteroids have non negligible probability only means the physics is sensitive to initial conditions or that the measurements are loose. (Both are true)

[MauranKilom]

Given everything you said is true, under those assumptions 3% of those asteroids that we identify as being in said 2km box will hit earth, unless the forward simulation is wrong (implausible) or the measurement error distribution is substantially wrong (also seems unlikely).

What your analysis is not touching on is the prior probability that an asteroid will hit earth (you collapse this to "any asteroid will either hit or not", but that is not helpful for "model calibration" or whatever you want to call this) - or, equivalently, the prior probability of making (a series of) observations with a certain uncertainty/error distribution. If that prior were actually as uniform as each measurement error suggests, I don't see any Bayesian wiggle room left for why we don't have those 3% of impact actually happen.

(I'm no expert, but presumably you need multiple measurements to predict a trajectory, and while their measurement error distributions may be independent, it seems plausible to me that the prior probability of making two specific noise-affected observations, i.e. of the asteroid being on a certain trajectory, is most likely not so uniform. That's the part that I'd like to learn more about though.)

[gyrovagueGeist]

I think some confusion here seems to come from the following interpretations:

-Then what does 3% mean? Surely it means "given the data we have, one in every 33 will hit" -Given everything you said is true, under those assumptions 3% of those asteroids that we identify as being in said 2km box will hit earth.

Both of these statements are false. The probability density is over our knowledge of the state variables/state space for this asteroid, not over asteroids. The hypothetical sample of asteroids is not drawn from the distribution I'm talking about.

Going back to the simplified example: With the uniform prior on the box, our probability means that 3% of the volume of this box would lead to an impact if an asteroid was centered at a point in that volume at this time of measurement.

It doesn't say anything about hypothetical realizations of this asteroid (it is not clear what this would be sampled from or what it means in a precise sense to repeat a 1 time event) and says even less about the sample of (nearly) independent asteroids observed in the past. The probability measure only describes the measurement uncertainty on properties of this particular asteroid. It is not conditioned on or related to statistics on impacts of "general asteroids".

But "presumably you need multiple measurements to predict a trajectory" and your notes about independence and uniformness being bad assumptions are absolutely correct tho. I agree 100%

My comment above is mostly an attempt to make a simple example to clarify what the probability measure being measured here is. It's not a physically realistic example :) and definitely doesn't make good assumptions about what information is needed and what error distributions that information would have! I don't do space and didn't want to make guesses

Calibration here would have to be over multiple measurements of the same asteroid (which my example doesn't touch on). Likely by predicting trajectories at different intervals and matching the likelihood of later observations.

Verifying multiple observations leading up to a 1 time event is a very different than, say, verification of simulations of an internal combustion engine design where measurements of a real world prototype can be conducted repeatedly and independently to learn/calibrate some fundamental properties or initial conditions like chemical kinetic coefficients and such.

For general interest/lectures/fun, the general field that studies how to push uncertainties forwards/backwards/calibrate a mathematical model and simulation is called "Uncertainty Quantification". Also not an expert lol, I was just surrounded by a bunch in my cohort