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> For example, the Ptolemaic system could approximate the movement of the planets to any degree if you added enough "wheels within wheels" but since these were

> For example, the Ptolemaic system could approximate the movement of the planets to any degree if you added enough "wheels within wheels" but since these were "the wrong wheels", the necessary wheels grew without bounds to achieve reasonable approximation over time. That would be an example of over-constraining your model (i.e. imposing the arbitrary constraint of a stationary Earth).

joe_the_user 13y ago
I don't think this is useful way to phrase the situation. A system of Ptolemaic circles can approximate the paths taken by any system. So the system really isn't absolutely constrained to follow or not follow any given path. You could claim you have constrained your model not be some other better model but that, again, seems like a poor way to phrase things since a more accurate model is also constrained not to be a poor model. Even specifically, the Newtonian/Keplerian system has the constrain of the sun being stationary as much as the Ptolemaic system has the constraint of the earth being stationary. Edit: As Eru points out, the Ptolemaic system basically uses the Fourier transform to represent paths. Thus the approximation is actually completely unconstrained in the space of paths, that is it can approximate anything. But by that token, the fact that it can approximate a given path explains nothing and the choices that are simple in this system are not necessarily the best choices for the given case, estimating planetary motion. See - http://en.wikipedia.org/wiki/Deferent_and_epicycle http://en.wikipedia.org/wiki/Deferent_and_epicycle
- rcthompson 13y ago
  That's a good point, but after re-reading tel's original comment, I think my statement is still correct. Notice that tel's statement was that "you can over-constrain your model space so that the true model cannot be found". This doesn't necessarily mean constraining your model so that the true model is excluded from your parameter space. If your constraints technically encompass the true solution but only admit an overly complex parametrization of the solution, then it will still reduce (perhaps drastically) your power to find the true model. In this case, "overly complex" means unnecessarily many nonzero (or not almost zero) coefficients in the Fourier series.
  - joe_the_user 13y ago
    My argument is that there are two kind of situations: * The model could encompass the behavior of the input in a smooth fashion if it's basic parameters are relaxed. * The model would tend to start finding models that are wildly different from the main model at the edges (space and time) if its parameter are relaxed, even if the model would eventually find the real model with enough input and training. one has to handle these two conditions differently, right?
Natsu 13y ago
> (i.e. imposing the arbitrary constraint of a stationary Earth). It's not really arbitrary--given the understanding at the time, there was no ability to measure the motion of the earth. In particular, stellar parallax which was understood as a contra-indication and too small to measure just yet. So a non-stationary Earth went against what they knew at the time rather strongly. That said, relativity comes back and makes choosing a frame of reference arbitrary in the end, though some are easier to do physics in than others.
- rcthompson 13y ago
  Yes, that's true, I'm intentionally ignoring history and calling it arbitrary from our modern perspective.

joe_the_user 13y ago

I don't think this is useful way to phrase the situation. A system of Ptolemaic circles can approximate the paths taken by any system. So the system really isn't absolutely constrained to follow or not follow any given path. You could claim you have constrained your model not be some other better model but that, again, seems like a poor way to phrase things since a more accurate model is also constrained not to be a poor model. Even specifically, the Newtonian/Keplerian system has the constrain of the sun being stationary as much as the Ptolemaic system has the constraint of the earth being stationary. Edit: As Eru points out, the Ptolemaic system basically uses the Fourier transform to represent paths. Thus the approximation is actually completely unconstrained in the space of paths, that is it can approximate anything. But by that token, the fact that it can approximate a given path explains nothing and the choices that are simple in this system are not necessarily the best choices for the given case, estimating planetary motion. See - http://en.wikipedia.org/wiki/Deferent_and_epicycle http://en.wikipedia.org/wiki/Deferent_and_epicycle

rcthompson 13y ago
That's a good point, but after re-reading tel's original comment, I think my statement is still correct. Notice that tel's statement was that "you can over-constrain your model space so that the true model cannot be found". This doesn't necessarily mean constraining your model so that the true model is excluded from your parameter space. If your constraints technically encompass the true solution but only admit an overly complex parametrization of the solution, then it will still reduce (perhaps drastically) your power to find the true model. In this case, "overly complex" means unnecessarily many nonzero (or not almost zero) coefficients in the Fourier series.
- joe_the_user 13y ago
  My argument is that there are two kind of situations: * The model could encompass the behavior of the input in a smooth fashion if it's basic parameters are relaxed. * The model would tend to start finding models that are wildly different from the main model at the edges (space and time) if its parameter are relaxed, even if the model would eventually find the real model with enough input and training. one has to handle these two conditions differently, right?

Natsu 13y ago

> (i.e. imposing the arbitrary constraint of a stationary Earth). It's not really arbitrary--given the understanding at the time, there was no ability to measure the motion of the earth. In particular, stellar parallax which was understood as a contra-indication and too small to measure just yet. So a non-stationary Earth went against what they knew at the time rather strongly. That said, relativity comes back and makes choosing a frame of reference arbitrary in the end, though some are easier to do physics in than others.

rcthompson 13y ago
Yes, that's true, I'm intentionally ignoring history and calling it arbitrary from our modern perspective.