Statistics on mathematical "data"?

Question

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 $x$).

Answer 1

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.

Answer 2

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:

• Interactive proof systems
• Probabilistically checkable proofs

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).

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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.