We do statistics on data which we assume to be outcome of random trials. The question however, is -- are we assuming that the outcome can be represented as a random variable?

The outcome of an experiment is representable as a random variable if that function is measurable. If the mapping from the event space, to the set of real numbers, is a measurable function, then we can even think of assigning a probability to that random variable.

Thought experiments are fine. We can assume that the outcome of a thought experiment, like rolling a dice, has an outcome that can be represented by a measurable function which we designate as the random variable.

When dealing with real data, don't we assume that the process that generates that data, can be represented as a random variable? Is that simply an assumption, or are there concrete reasons/tests to believe that it is indeed measurable?

  • 4
    $\begingroup$ Are you confusing "measurable" in colloquial with measurable mathematically? Measurable is a mathematical definition and not your belief. $\endgroup$
    – user318514
    Commented Jun 2, 2022 at 19:45
  • 3
    $\begingroup$ Data generation "processes" aren't measurable, but random variables are. Measurability becomes an issue when modeling stochastic processes in continuous time, where it is related to what kinds of variables can be well-defined and given meaningful properties. As in any field of applied mathematics, a model is used to reason about the data and a theory. There no reason to put it on some elevated ontological plane to be worshiped as "real" or denigrated as "simply an assumption." $\endgroup$
    – whuber
    Commented Jun 2, 2022 at 21:22
  • $\begingroup$ @whuber All I wanted to ask was how do we even assume that a certain process has a probability distribution? A random variable is a "measurable function" from the sample space to a measurable space. I am aware that there are tests to check if the data is normally distributed (the normality tests) but how do you even test for something as basic as whether the given data set can be represented as realizations of a "random variable"? For all we know, the mapping from an event space to a measurable space, if it exists for that process, may not be a measurable function right? $\endgroup$ Commented Jun 3, 2022 at 3:52
  • 3
    $\begingroup$ The map is not the territory and the model is not the reality. The formal construct of observations as realisations of a random variable is used for prediction or explanation purposes, it does not imply a belief that it is real. Take for instance the SIER modelling of COVID propagation, a useful if artificial representation of the real thing. $\endgroup$
    – Xi'an
    Commented Jun 3, 2022 at 6:49
  • 1
    $\begingroup$ "Bernardo and Smith (2000) define M-closed problems as those for which a true model can be identified and written down but is one amongst finitely many models from which an analyst has to choose. By contrast, M-complete problems are those in which a true model (sometimes called a belief model) exists but is inaccessible in the sense that even though it can be conceptualized it cannot be written down or at least cannot be used directly. Effectively this means that other surrogate models must be identified and used for inferential purposes." Le & Clarke (2017) $\endgroup$
    – Xi'an
    Commented Jun 3, 2022 at 8:28


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.