Lets say there is a theoretical relationship you want to prove over all values of some variable. For example, $F(x)$ increases with $x$.

You are unable to come up with a general theoretical proof. However, you can calculate $F(x)$ for any specific value of $x$, and the expected relationship holds.

Presumably, you can then run some sort of regression on the "data" provided by a whole bunch of the values you specifically calculated. (using $x$ as the IV and $F(x)$ as the DV). Then, you can show that an increasing relationship is supported by the "data" and try to come up with a best fit, etc.

This is sort of like "comparative statics" only taking it one step further -- trying to prove the direction (even shape) of a relationship, rather than just the ordinal relationship of two values.

Is there a name for this type of process -- running statistics on mathematical solutions, vs. measured data? Is this a convincing way to argue a theoretical point? Isasmuch as running an experiment and collecting data values and running statistics on that is?

The final goal is not a theoretical contribution on $F$'s movement with $x$, but rather a model that provides a useful takeaway (e.g. since $F(x)$ increases with $x$, it is advisable to try to minimize $x$).

  • $\begingroup$ Why don't you tell us what f(x) is. $\endgroup$ – Michael Bishop Oct 15 '11 at 23:00
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    $\begingroup$ @Michael In this case, it is explained in this post: stats.stackexchange.com/questions/17022/… I am trying to come up with an alternate plan if I am unable to solve the relationship mathematically. $\endgroup$ – OctaviaQ Oct 16 '11 at 1:21

Any time you do computer simulations to evaluate the performance of a statistical method (eg, power), you are approximating a calculation that might conceivably be calculated analytically (power is a probability). You could also conceive of doing exact-ish numerical calculations: summing exhaustively across all possible outcomes.

In a paper to appear in Genetics, I wrote:

Simulations are most flexible and are generally simpler to obtain, but lack precision. Numeric calculations can be precise, but can be computationally intensive. Symbolic results are more general than numeric calculations, can enable quicker calculations in software, and have the potential to provide more clear insight.

I was trying to justify some crazy efforts I put into some analytical calulations that could easily be obtained numerically.

So regarding your question, I think numeric results can be quite compelling—not real proof, but likely sufficient to make the point in the range of parameter values considered. Subjecting the results of numerical calculations to regression analysis can be really useful. (I've done that to figure out and/or verify an answer that ultimately I derived analytically.) But again it's not real proof, though if correct to within round-off error it would be pretty compelling just not completely satisfying.

The advantage of numerical calculations over simulations is that you can get to $10^{-13}$ error whereas for simulations you'd never get that precise.

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    $\begingroup$ +1 While I love simulation as a method, it's good to know exactly what you're doing. $\endgroup$ – Fomite Oct 15 '11 at 23:34
  • $\begingroup$ @Karl... hmm.. so if I understand correctly: Simulation is using, say, a monte carlo method to approximate a value. Numerical analysis is using a computer program to approximate the value of a mathematical formula/integral. And statistical analysis can be run on the results of either method. Is that much correct? What would I call running statistics on the purely analytical solutions at various points? $\endgroup$ – OctaviaQ Oct 16 '11 at 0:47
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    $\begingroup$ @Jand - yes what you're doing is numerical calculations; I just thought it might be helpful for you to take it a bit further and think of simulations. $\endgroup$ – Karl Oct 16 '11 at 0:47
  • $\begingroup$ If I can use Integrate instead of NIntegrate to get exact values for all the points I expressly use, then it's not simulation or numerical methods... it's just finding a statistical relationship among mathematical points. Or is that simulation? What would I call that? $\endgroup$ – OctaviaQ Oct 16 '11 at 0:49
  • $\begingroup$ @Karl It is helpful, thank you!! I actually did a simulation first. I was told I should try for an analytical solution. I've been working on it for months. And I'm just not sure if what I have (assuming I use Integrate instead of NIntegrate... just get INSANELY long fractions) counts as analytical... I am not sure what category to put it into. $\endgroup$ – OctaviaQ Oct 16 '11 at 0:50

This is akin (Update: see below) to methods used in automated probabilistic proof systems or cryptographic verification systems. An example would be a zero knowledge proof.

Several other areas to explore:

In general, one does not try to do statistical analyses on the relationships found over the the input and the response (i.e. ${x}$ and ${F(x}$) spaces, though these may be analyzed probabilistically over finite spaces. Infinite spaces are a bit more of a challenge, though you may be able to construct some mappings to finite classes (i.e. subspaces that may be infinite, but where a solution in a subspace is representative for all points in the subspace, even if this is only the case with some probability).

Update 1: This is akin to such methods, with the caveat that such methods are usually applied in the context of someone claiming to have a proof, as opposed to probabilistically reasoning about a proof. The distinction is important. Such reasoning cannot generally be accepted as a proof, so one could say it is like probabilistic conjecture verification. This is more like the numerical verifications done for the Goldbach Conjecture or the Riemann Hypothesis.

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