Correlation not = Causation, but if No Correlation, No Causal Relation Exists? True or No?

Question

I understand that correlation does not imply a causal relationship. But, if two things (for example Fed Reserves actions and Inflation) are not correlated, does that imply that there can be no causal relationship?

Answer 1

Unfortunately no. There could be a causation even if there is no correlation. Correlation is related to ability to present data related to two, basically, objects in a compressed form, so it is possible to predict the relationship between two our more objects in its entirety. When we say $y=x^2$, it is not important what x or y are, the correlation is saying whenever $x$ has any value $y$ will have a specific other value.

There are two reasons that correlation is not related to causation:

1. We do know only about correlation and causation within a limited time and space frame, because we, examinators, are short-lived on this planet and some processes cannot be explained through any observation done from any given point in time. We do believe that we can understand the Universe by observing it only from our planet or Solar system, but this is very likely not so. It is though true that we cannot collect a sufficient amount of information about majority of processes that is happening in the Universe in any capacity
2. Some processes are correlated by simply being correlated in uncompressible way. Here we are not talking about chaotic systems, where we can always extract some information about the system, for example total energy. We are talking about unpredictable information all together that still has a physical interpretation (for example a probability that any given program will halt). Not only that such processes are deterministic, they are totally defined, yet we cannot find their values. You can try to find correlation experimentally or theoretically, but causality is more or less direct. It is just one unknown number.

So your question is about describing the world around us. Is all causality contained in correlation? No. It is not only that we cannot know everything, we cannot even start knowing some bits of everything. And we can never know which part we can or which part we cannot know.

So what we do: we keep on discovering as much as we can and that is it. Believing that only if we continue this process long enough, we will start learning just about everything that there is in this world, is futile.

It is even worse than it seems. Most causality do not have any compressible correlation attached to it. However, since we are biologic creatures we are concentrated on those correlation/causation that can save us energy, i.e. that are compressible and that we can discover and use to save energy. That is what life is about. We are not here for nothing.

Well, maybe humans are not exactly behaving like that, but the entire biology is about that. We... we are dreaming a little bit more than what biology allows.

Answer 2

I think this question makes sense and can be clearly answered, if you focus just on the mathematics. To avoid difficult philosophical questions, I define causal relationship in a strictly mathematical sense as functional relationship, i.e.
$$ X \text{ causes } Y \text{ if there exists a function } f \text { such that } Y = f(X). $$

Of course other definitions of causality may be possible and meaningful but it seems to me this is a very strong interpretation of causal relationship.

Having now a proper definition it is easy to give an example of two uncorrelated random variables $X$ and $Y$ with $Y=f(X).$

Let $\Omega = \{-1, 0, 1\}$ with probability $\mathbb{P}(\omega)=\frac 1 3$ for $\omega \in \Omega.$ The random variable $X$ is defined as $X(\omega)=\omega$ and $Y$ defined as $Y(-1)=Y(1)=0$ and $Y(0)=1.$ Notice that $Y$ is indeed a function of $X$.

Since $XY=0$ you have $\mathbb{E}[XY]=0$ and from $\mathbb{E}[X]=0$ you can conclude $\mathbb{E}[X]\mathbb{E}[Y]=0$. This means the covariance $\mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y]$ and also the correlation of the two variables is zero.

The take home message beyond this simple counter example is: Correlation is only a very limited way to describe the relationship between random variables.