Probabilities are a powerful tool to model reality. And while they are
interesting on their own, they are more interesting in a situations where a
condition C leads to a result R.
Sadly, causation has different meanings in different contexts. And it’s important to agree on which model of causation is used in any conversation.
Mathematical Certainty#
In the simple sense, if something causes something else, we can use Logical Implication to capture part of this.
If C holds, then R must also hold.
\(C \implies R\)
And if the condition can be applied to single elements of some set, it holds
that, for every x where C is true, R is also true:
\(\forall x. C(x) \implies R(x)\)
And while this works reasonably well for mathematics and some processes in reality, it generally fails when a statement is made about a natural process.
The Problem#
A rather simple example of how the mathematical model fails once we consider reality is mortality of some incidents.
In the mathematical context, disproving a causality is simply done by finding a
counter example. I.e. there is an x for which C holds, but R does not:
\(\exists x. C(x) \land \lnot R(x)\)
.
In the example, this means, if there’s even a single person that survived e.g. a car crash, then car crashes do not cause death.
Statistical Relevance#
Thus, in practice we have a concept of Statistical Relevance that allows to make statements of causality with a bit of uncertainty. I.e. when the causality holds in the vast majority of cases, the statement remains true in the presence of counter examples.
Assuming a \(p \leq 0.05\) this a statement is true when it holds for 95% of cases: \(C(x) \implies ℙ(R(x)) > 0.95\)
The article on human on the simple english wikipedia has a statement:
Humans are bipedal, which means they walk on two legs.
And while we are aware of situations where this isn’t true (babys, old people, various physical disabilities), it’s true often enough that it can be said as is.
Even more Problems#
Statistical relevance allows to make statements about systems with occasional outliers. I.e. when something is generally true, but some modality that’s unknown or hard to model can rarely cause an exception.
And while that’s quite powerful, it comes with its own problems.
Highly Multi Causal Systems#
This form of causality is still largely like the certainty causality. It makes statements about things which are mostly \(\forall x. C(x) \implies R(x)\) but allows for some tiny deviation.
Though if we look at e.g. health effects of smoking it’s clear that this model is not enough.
A 22.1% (or even worse 11.9%) probability doesn’t meet the mark for significance by a long shot. But we are still confident that smoking does cause cancer. So we need a better model of causality that allows for this.
Erasure of Outliers#
While the ability to talk about things that hold “often enough to not worry about outliers”, it does fully erase the outliers. With the 95% mark as cutoff, this can be a quite large portion.
E.g. the current life expectancy in Germany is 82.20 years. Fertility rate (As of 2020) is down to 1.59. A pregnancy lasts 9 months, so 3/4th of a year.
So we can roughly say that the chance of a woman currently being pregnant is: \(\frac{0.9*1.59}{82.2}\approx0.017\) .
I.e. we can say with 98% certainty that no woman is pregnant. Or even with 99% that nobody is pregnant. Which is obviously not great.
Probalistic Causality#
To properly model the sort of causation underlying e.g. carcinogens, we need to dive deeper into probabilities.
The simple system under this is called
Conditional Probability.
I.e. the probability that something happens when the condition C is fulfilled.
The result R is more likely when the condition C is given, than it is
without this restriction
\(ℙ(R \mid C) \gt ℙ(R)\)
.
So if we plug in the numbers from before, we get: \(ℙ(Cancer \mid smoking) = 11.9% \gt 1.1% ℙ(Cancer)\) . Which shows that does help understanding the health impact of smoking.
There’s always issues#
While I’m not aware of any causalities that this model cannot represent, it has other issues.
With the rather relaxed requirement it is easy to confuse correlation with causation. Or even reverse the cause and effect relationship.
The spurious correlations website is based on this idea. Something that superficially (mathematically) looks related or even causal but isn’t.