At its simplest, the forward problem is like a functional composition:
y = F( G( H(x) ) )
@gliese1337 nearby, in an excellent comment, addresses the more complex inverse problem: when you have outputs (y's above) from such a process, how do you determine the most likely model (F, G, H) that produced it?
So for example, if you have a bunch of crater ejecta, what can you tell me about the mass of the incoming objects?
- Start with a value + divergence and then propagate that state through your transforms to get a mean + divergent final state.
I should note that in the given example the base divergences need to be found through some method, which is usually running a Monte-carlo around some baseline state, and then making the significant assumption that new calculations (with small deltas) follow roughly the same statistical behavior.
For the backwards case, that can be a significantly more challenging case, as it basically comes down to performing approximate eigenvalue decomposition. However, there are a lot of SOTA tools like pretty much anything from this list:
- http://en.wikipedia.org/wiki/Pattern_recognition#Algorithms
IE, the crater example is fairly close to the setup for a Hidden Markov Model, where we get final data, and assume that it is based on a set of hidden states. Or for lighter reading, almost any of the major tools used in the Netflix recommendation contest, which is effectively an enormous version of this problem.
Hmm, not in general. (I've been working on various problem-specific versions of the backwards case for most of my career.)
Disclaimer: my lab mate has been organizing these workshops for the past couple of years and I'm currently procrastinating from preparing a presentation on this material for a reading group later this week, so I've presently got mixed feelings towards it.
In one of my lives, I've calculated ballistics. When calculating ballistics, you can probably think of a lot of factors that might go into it.
- Release conditions (velocities, angles, ect...)
- Environmental / trajectory conditions (winds, gusts, different media, ect...)
- Impact conditions and behaviour (spalling, rebounding, fragmentation, yatta....)
For more reading, you can look at something like: http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/2007002... (only uses probabilistic methods for transport)
Anyhow, one way to attempt modelling of this is to run a Monte-carlo of the parameter space. Effectively, you run all the permutations of release, all the permutations of those permutations for transport, and then all the permutations^3 of the impact variation.
Now, one of the advances that has come about from efforts like that, is that probabilistic models for some of these steps have been created to approximate the results in a time effective manner.
For example, in the above, they note that instead of running a Monte-carlo of trajectories and releases, a single trajectory is instead run, and then a probabilistic flight cone is created which calculates parameters based on standard deviation from the mean.
A general, and robust version of a language like this would allow you to specify all of your parameters with similar types of effects. IE, you would have mean and then deviation or statistical curve envelopes you would provide around that mean which would be propagated through calculations, and spit out an overall convoluted probability at the end of the pipeline.
In ballistics for example, you might have a mean release condition, with statistical variation on release. That would then be fed into a trajectory calculation, which would do so based on a mean, but with a transport cone that does statistical variation on the mean flight. Finally, impact might be calculated around a mean value, but would then carry the statistical variation through the release and flight. At the end, you would then be able to pull a probability number, as well as impact condition at that probability from any point in the potential impact zone.
At least that's how I see the use of probabilistic languages.