What does one mean by a 'random variable' in the real world?

Question

I want to understand what it means for process to be random or for an object to have random properties.

Contrary to what I used to believe and to popular belief J Schmidhuber argues that true randomness is still a hypothesis.

So random appears to be related with information we do not know in advance either because it is too difficult to compute or is unavailable at that moment, but is not a property inherent to the process or object.

I've been searching for an explanation to this and so far I have found algorithmic randomness, which explains more formally what we mean by random sequence in mathematical terms and shows certain properties that we would like a random sequence to have, but as far as I can tell this is not related directly to real life randomness.

As a thought experiment suppose we have a 6-sided die which we have not yet physically measured. A roll might be appear to be random at first, but that is because we do not know enough about the die or the experiment itself (properties of the thrower and the environment it rolls in). I have not found an experiment where this is not the case.

Related question

Answer 1

After studying statistics for a reasonable amount of time, my personal view of randomness is the distribution of some metric conditioning on a set of information. The randomness is a function of how much we know about this particular instance of the metric.

Let's take the classic six sized die example. If I'm about to roll it, one could make an argument that given all the forces acting on that die, the outcome is deterministic. We can imagine that if we understood and could perfectly control all these forces, we could make a machine that always gives us the roll we want. It's just a lot of physics.

But those are a lot of forces I don't understand. So I'm just going to condition on the fact that the die is fair (I assume) and average over all those unknown forces. In that case, we can think of an near infinite set of possible forces acting on this die, and we believe in 1/6 cases, a 1 shows up. Thus, if all I know is that the next roll is going to be from a fair die but I don't know all the acting physics on that die, the best I can do is say that the outcome belongs to a set in which 1/6 of the outcomes are 1, so I say to the best of my knowledge, P(roll = 1) = 1/6. I'm essentially averaging over this unknown forces that would have made my metric deterministic had I known them.

As another example, suppose some asks "what's the distribution of heights of 23 year old men?". To be honest, I'm not exactly sure the answer to this, but if someone told me that it was approximately normally distributed with mean 5'8" and standard deviation 4" that would seem believable to me. But note that if I provided you with more information, i.e., "what's the distribution of heights of 23 year old men who's father was 6'1"?", I've provided you with more information that allows you to narrow down the larger set (23 year old men) to a smaller subset (23 year old men with fathers 6'1"). Mostly likely there is less variability in this subset which then reduces the variability of the metric in this subset. Thus we see that the variability of a particular metric (height) is very dependent on exactly what we are conditioning on having known (age? gender? height of father?).

Answer 2

Interesting thought question.

I view the concept of true randomness in a negative sense. For example, lack of order, predictability or, for a spherical ball, lack of imperfections.

If a rolling ball is not perfect (as in round, perfectly smooth,...), one may be able to predict with higher probability, where on the ball, it will come to rest on. With a perfect theoretical ball, all points are equally likely.

So perhaps, per J Schmidhuber, 'true randomness is still a hypothesis'.

Answer 3

Maybe another philosophical point to add is the following: In mathematics we have a very clear and precise understanding what a random variable is in these days. I always imagine it as follows: The basic set people often talk about is $\Omega$. Every $\omega \in \Omega$ is one 'configuration' containing all possible information of a universe we might live in. A random variable $X$ is a map (with certain properties) from $\Omega$ to the space of possible outcomes... let's say we talk about a dice then $X : \Omega \to \{1,2,3,4,5,6\}$. I always imagine that as an instance of the Laplace daemon: https://en.wikipedia.org/wiki/Laplace%27s_demon: Given all information about the current temperature in the room, the initial angle and position of the dice, the mood of the person that rolled the dice, ... (i.e. really ALL information available encoded in $\omega$) we can perfectly predict the landing position of the dice because it is merely physics and $X$ is that deterministic Laplace daemon.

Of course, philosophically one could say: Is that really how the universe works? Doesn't quantum mechanics and the Planck universe tell us that we cannot deterministically understand the universe with all the mathematics and physics that we have nowadays (according to what I understood, the Planck universe tells us that there is a minimal unit of space and if we go below that size then essentially we cannot do physics anymore)? Yes, it does. Therefore the question whether or not that is a good definition of random variable is still 'open' I would say.

If it is still unclear whether that is a good definition or not, why are we using it so often for the theory behind the models we produce? The only reason here is: It seems to work well in practice. It's like in physics: Shouldn't we throw away all physics that we know if we can see that we can never fully understand the universe with these methods? Well, we could certainly do so but then we would not have fancy features like GPS ... so we rather keep the imperfect version and make the best out of it.

How is that all related to reality? Let say that we have collected some dataset $(x_i, y_i)_{i=1,...,n}$. Why do we make assumptions that these come from random variables in the first place? Sometimes we want to prove that a certain model is best suited for this dataset and in order to do that we need to have some ground rules set. However,

we can never really verify even whether the $x_i, y_i$ come from random variables $X_i, Y_i$ in the mathematical sense above, let alone whether they satisfy some (e.g. linear) relationship!!!

So in reality, even if we have true randomness, we can never really be sure about it. Is that a shitty way of doing data science? Well yes and no: Again, it is a way that is not perfect but we use it nevertheless because we can earn a lot of money with it (and make things better for the world and come up with nice things like google internet search, etc).