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Comment by VierScar | Hacker News Reader

I'm sure there's a reason, but it seems like an unusual use of Monte Carlo - it's all deterministic and there is no opposing player making choices. Must have something to do with uncertainties in projected orbits or imperfect simulations maybe?

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>it's all deterministic and there is no opposing player making choices

It's not deterministic, it's chaotic. That is the nature of the N-body problem. We can only approximate trajectories in such a system using numerical methods, within a certain margin of error. In principle, the object is gravitationally interacting with everything else in the solar system. But for the most part, most interactions are negligible and could be ignored (eg, other small objects far away), except of the large bodies. But there are many unknowns (as stated before), where initial conditions will affect the outcome of the trajectory simulation, and errors will certainly amplify over time. I'm guessing Monte Carlo is used to "fuzz" the simulations with randomised initial conditions to account for the range of unknowns, and see what the outcome is under these different scenarios.

At the very least you can use monte carlo to provide examples from the expected distribution. It's hard to visualize a probability cloud.

Chaotic doesn't mean non-deterministic, it just means that small changes in initial conditions result in a large change in the trajectory. The system itself can be both chaotic and deterministic.

It's also a reasonable question to ask, because the simulations are deterministic. It's just that because the system is also chaotic and there's noise in the measurement, that can result in a large spread of deterministic trajectory simulations.

It's only deterministic in the sense of the mathematical constructs that models the system, like differential equations that drive the simulations at each finite time step. But the information or the state which the simulation is applied on is always chaotic. That is because delta at each time step is an approximation with some error. It's impossible to make the state in the system behave deterministically, because that requires time deltas to approach to zero (or infinite amount of infinitely small differential steps).

Energy drift doesn't make the system non-deterministic, it just means that the time evolution has some error. The error is still deterministic. If you simulate a deterministic but chaotic system like n-body orbitals with a non-symplectic integrator, you'll always get the same result for the same initial conditions. The drift created by the finite timestep will also be the same.

It’s the error with the ground truth that you can’t predict. Otherwise you would just be able to cancel it out. You can only predict probability distributions..

If you're saying that it's the uncertainty in the initial measurement, then we're in agreement. If the initial measurement were perfect, the only source of error would be the finite timestep. N-body simulation itself is deterministic, and so the only source of randomness is our uncertainty about the object's true mass, size, shape, position, velocity, etc.

The N-body _reality_ _might_ be deterministic. The N-body simulation using digital computers will technically still introduce errors because of the time steps even if you had perfect knowledge of initial conditions.

The errors are deterministic. Determinism has nothing to do with the existence of errors, it's about uncertainty. They're different things. A system that is deterministic will produce the same results every time given the same initial conditions. If there are numerical errors, they will be identical for each run. A non-deterministic system will give you different results every time given the same initial conditions, with some variance. You can still have numerical errors in such a system.

Ironically, reality probably isn't deterministic. It definitely isn't at small scales (e.g. radioactive decay). If it's non-deterministic at a macro scale, the effect is small enough that we don't see it.

We don't know the configutation it's in precisely. We don't know the initial conditions. Small unobservable differences will lead to large difference in outcome. That's the chaotic part.

I get that. I'm pointing out that these are separate factors. Chaotic does not imply non-deterministic, and vice versa. The only source of randomness here is the uncertainty in the observation of the object, because (as you point out) multiple combinations of parameters could produce the same observation, and each one will have a different trajectory. The randomness doesn't come from the chaotic nature of the system, it comes from noise in our measurements. It also doesn't (as other posts are claiming) come from energy drift in the simulation, because that's also deterministic.

The observations are not 100% certain. There are a variety of body states and configurations that might result in the same (few) pixels being lit up in the few measurements collected so far. As additional measurements are collected, some possibilities may be eliminated and the uncertainty of the trajectory can be reduced. This usually results in the impact probability converging toward 0%.

...or 100%. But yeah, the MC comes in this way. You have a current most probable value for the position and some distribution around it, depending on the precision of the measurement device etc. That can be a high-dimensional space. You draw some (many) random points from this space and propagate them all deterministically. Taking into account how likely a certain random point was in the first place, you can then estimate the hit probability.

MC is numerically approximating an integral. Here it replaces the high-dim integral over the start parameters.

Looks like alpha zero stole the original meaning of Monte Carlo… it is not required to have players, or a game, to use the method: https://en.m.wikipedia.org/wiki/Monte_Carlo_method

It wasn't stolen, it was a well-established particular case well before alphago

https://en.wikipedia.org/wiki/Monte_Carlo_tree_search

I only meant the total mind share :) looks like folks have never heard of MC in other contexts than game bots.

I would assume that it is because we have imperfect knowledge of the state of the asteroid (i.e. mass and current position/velocity/...). This imperfect knowledge is characterised by a probability distribution. Similarly the state of all other objects in the solar system is only known up to some distribution. To propagate the information forward in time to impact requires a complicated function f(state of solar system; state of asteroid). If all of the data was known (and expressible numerically) with perfect accuracy, and f were computable with perfect accuracy then all would be good. But as noted, (state of solar system; state of asteroid) is a probability distribution, and there are very few distributions and very few types of maps f that are amenable to analytic transformation. For example if the state was a normal distribution with mean x and covariance P, and f were a linear transformation, then x,P mapped through f is also normally distributed with mean y and covariance P_y, you can get the mean of the transform as y=fx, and P_y = fPf' (where ' indicates transpose). Needless to say our knowledge of the state of the asteroid and the solar system is probably a rather complicated distribution, and the n-body problem is not a linear transformation. Monte-carlo simulation is often used to propagate probability distributions through non-linear transformations.

It is very common to use Monte Carlo for deterministic problems. It's just an integral over complicated PDFs

It's "deterministic" the same way the weather is.

Exactly. It will do exactly this, as long as nothing changes.

Right:

- "Since we saw it so briefly, our knowledge of its orbit is not that great"

- "[for example, in 2016 the data shows] a large chunk of sky where it could have been, and [the object is quite small."

- "Our knowledge of the diameter of this object is a bit fuzzy, because of surface reflectivity,"

The guesswork is uncertainity about the object's exact paraneters. Because of this they have to use informed estimates (scientiffic guesswork).

Uncertain observations, I'm sure

Right - we only got a short glimpse of it, not enough to get a high confidence of its trajectory.

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