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The tag excerpt of says:

A formalization of relationships between stochastically (randomly) related variables in the form of mathematical equations.

This seems questionable to me. I guess cases where this holds exist, but what about the majority of the relatively simple cases? The basic models often have deterministic relationships between variables; e.g. in a linear model, $$ Y=\beta_0+\beta_1 X+\varepsilon. $$ $Y$ and $\varepsilon$ are random variables, $X$ may be random or nonrandom, so there are objects that are random. However, the relationship between these objects (here in the form of the linear model) is fixed. Does it make sense to say variables are stochastically (randomly) related here?

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  • $\begingroup$ You could also object that $Y$ and $X$ are deterministic in most applied settings... $\endgroup$
    – Xi'an
    Commented May 12, 2021 at 13:02
  • $\begingroup$ Not sure I understand. I wrote explicitly the opposite in my last paragraph. @Xi'an $\endgroup$
    – Richard Hardy
    Commented May 12, 2021 at 13:05
  • $\begingroup$ Yes, I understand what you wrote but a similar objection could be made on the opposite basis. $\endgroup$
    – Xi'an
    Commented May 12, 2021 at 13:09
  • $\begingroup$ Then I probably misunderstood you. Are you saying that one could make the case that $Y$ and $X$ are deterministic rather than stochastic? And if this were technically possible, would this not go against most of what statisticians do? To clarify, I am focusing on the tag excerpt in the particular context of statistics and Cross Validated. $\endgroup$
    – Richard Hardy
    Commented May 12, 2021 at 13:21
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    $\begingroup$ In this post, I don't see any counterexamples to the characterization. Any deterministic value can always be viewed as a degenerate version of a random variable, so every non-stochastic (quantitative) model can be viewed as a special case of this definition. Moreover, your question hinges on changing the meaning of "related" to exclude the effect of $\varepsilon$ in the linear model. $\endgroup$
    – whuber ♦
    Commented May 12, 2021 at 15:36

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The definition is far too narrow. To identify a model in terms of random variables is to put noise on a pedestal AND reduce all modelling to one set of applications. Not all systems are stochastic, for example, the equations that model the position of a swinging pendulum may have some small noise of regression fit for their application to any particular problem, but those equations are largely deterministic, and the "noise" may be one part in one million as a residual function. Worse, one man's noise for a deterministic model may be another man's data for a more exact model. Suppose on a hot day someone puts an oscillatory fan blowing air in the same room as an exposed pendulum. Then, the "noise," i.e., the residuals of fit, that result from timed positional data for the pendulum may themselves be largely deterministic and subject to secondary deterministic modelling.

Or try another typical measurement problem, how long is a day?

Exact Day Length* — Sun, May 23, 2021

Today's prediction: 24 hours, 0 minutes, 0.0007688 seconds (0.7688 milliseconds)

The simple answer is 24 hours plus "noise." However, as above, the noise is predictable and the problem is deterministic to within a ridiculously small error. That is modelling at its best. Thus, to define what a model is and ignore the physical world (literally) is disciplinary myopia to coin a phrase.

To wit, I propose to say that a model is "The mathematical equation(s) formalizing the relationship(s) between variables."

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  • $\begingroup$ Thank you for your perspective. $\endgroup$
    – Richard Hardy
    Commented May 24, 2021 at 16:24
  • $\begingroup$ @RichardHardy Actually, thank you for your question. Things like statistical mechanics provide some of the most elegant statistical models. Indeed, something to be proud of. $\endgroup$
    – Carl
    Commented May 25, 2021 at 8:08

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